Haïm Brézis

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Functional Analysis, Sobolev Spaces and Partial Differential Equations

2010-11-02

Haïm Brézis

Positive solutions of nonlinear elliptic equations involving critical sobolev exponents

1983-07-01

Haïm Brézis , Louis Nirenberg

Operateurs Maximaux Monotones - Et Semi-Groupes De Contractions Dans Les Espaces De Hilbert

1973-01-01

Haïm Brézis

A relation between pointwise convergence of functions and convergence of functionals

1983-07-01

Haïm Brézis , Élliott H. Lieb

We show that if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-brace f Subscript n Baseline right-brace"> <mml:semantics> <mml:mrow> <mml:mo>{</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> </mml:mrow> <mml:mo>}</mml:mo> </mml:mrow> <mml:annotation … We show that if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-brace f Subscript n Baseline right-brace"> <mml:semantics> <mml:mrow> <mml:mo>{</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> </mml:mrow> <mml:mo>}</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\left \{ {{f_n}} \right \}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a sequence of uniformly <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Superscript p"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{L^p}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-bounded functions on a measure space, and if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f Subscript n Baseline right-arrow f"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> <mml:mo stretchy="false">→</mml:mo> <mml:mi>f</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">{f_n} \to f</mml:annotation> </mml:semantics> </mml:math> </inline-formula> pointwise a.e., then <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="limit Underscript n right-arrow normal infinity Endscripts left-brace double-vertical-bar f Subscript n Baseline double-vertical-bar Subscript p Superscript p Baseline minus double-vertical-bar f Subscript n Baseline minus f double-vertical-bar Subscript p Superscript p Baseline right-brace equals double-vertical-bar f double-vertical-bar Subscript p Superscript p"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:munder> <mml:mo movablelimits="true" form="prefix">lim</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> <mml:mo stretchy="false">→</mml:mo> <mml:mi mathvariant="normal">∞</mml:mi> </mml:mrow> </mml:munder> </mml:mrow> <mml:mrow> <mml:mo>{</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msubsup> <mml:mrow> <mml:mo symmetric="true">‖</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> </mml:mrow> <mml:mo symmetric="true">‖</mml:mo> </mml:mrow> <mml:mi>p</mml:mi> <mml:mi>p</mml:mi> </mml:msubsup> <mml:mo>−</mml:mo> <mml:msubsup> <mml:mrow> <mml:mo symmetric="true">‖</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> <mml:mo>−</mml:mo> <mml:mi>f</mml:mi> </mml:mrow> <mml:mo symmetric="true">‖</mml:mo> </mml:mrow> <mml:mi>p</mml:mi> <mml:mi>p</mml:mi> </mml:msubsup> </mml:mrow> <mml:mo>}</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:msubsup> <mml:mrow> <mml:mo symmetric="true">‖</mml:mo> <mml:mi>f</mml:mi> <mml:mo symmetric="true">‖</mml:mo> </mml:mrow> <mml:mi>p</mml:mi> <mml:mi>p</mml:mi> </mml:msubsup> </mml:mrow> <mml:annotation encoding="application/x-tex">{\lim _{n \to \infty }}\left \{ {\left \| {{f_n}} \right \|_p^p - \left \| {{f_n} - f} \right \|_p^p} \right \} = \left \| f \right \|_p^p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for all <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0 greater-than p greater-than normal infinity"> <mml:semantics> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>></mml:mo> <mml:mi>p</mml:mi> <mml:mo>></mml:mo> <mml:mi mathvariant="normal">∞</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">0 > p > \infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. This result is also generalized in Theorem 2 to some functionals other than the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Superscript p"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{L^p}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> norm, namely <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral StartAbsoluteValue j left-parenthesis f Subscript n Baseline right-parenthesis minus j left-parenthesis f Subscript n Baseline minus f right-parenthesis minus j left-parenthesis f right-parenthesis EndAbsoluteValue right-arrow 0"> <mml:semantics> <mml:mrow> <mml:mo>∫</mml:mo> <mml:mrow> <mml:mo>|</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>j</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> <mml:mo>−</mml:mo> <mml:mi>j</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> <mml:mo>−</mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>−</mml:mo> <mml:mi>j</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:mo>|</mml:mo> </mml:mrow> <mml:mo stretchy="false">→</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\int \left | {j({f_n}) - j({f_n} - f) - j(f)} \right | \to 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for suitable <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="j colon bold upper C right-arrow bold upper C"> <mml:semantics> <mml:mrow> <mml:mi>j</mml:mi> <mml:mo>:</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">C</mml:mi> </mml:mrow> </mml:mrow> <mml:mo stretchy="false">→</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">C</mml:mi> </mml:mrow> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">j:{\mathbf {C}} \to {\mathbf {C}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and a suitable sequence <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-brace f Subscript n Baseline right-brace"> <mml:semantics> <mml:mrow> <mml:mo>{</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> </mml:mrow> <mml:mo>}</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\left \{ {{f_n}} \right \}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. A brief discussion is given of the usefulness of this result in variational problems.

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A Relation Between Pointwise Convergence of Functions and Convergence of Functionals

1983-07-01

Haïm Brézis , Élliott H. Lieb

Combined Effects of Concave and Convex Nonlinearities in Some Elliptic Problems

1994-06-01

Antonio Ambrosetti , Haïm Brézis , G. Cerami

Équations et inéquations non linéaires dans les espaces vectoriels en dualité

1968-01-01

Haïm Brézis

On introduit dans le cadre des espaces vectoriels en dualité, deux vastes classes d’opérateurs non linéaires les opérateurs de type M et les opérateurs pseudo-monotones. On met en évidence plusieurs … On introduit dans le cadre des espaces vectoriels en dualité, deux vastes classes d’opérateurs non linéaires les opérateurs de type M et les opérateurs pseudo-monotones. On met en évidence plusieurs de leurs propriétés analogues à celles des opérateurs monotones ; en particulier, on résoud pour ces opérateurs des problèmes abstraits de type elliptique et parabolique, des équations intégrales, des inéquations variationnelles stationnaires et d’évolution. Suivent quelques applications.

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Semi-linear second-order elliptic equations in L1

1973-10-01

Haïm Brézis , Walter A. Strauss

a paper which is yet to be completed, we have obtained some analogous results for parabolic equations.\S 1.An abstract formulation of the monotone case.Let $\beta$ be a maximal monotone graph … a paper which is yet to be completed, we have obtained some analogous results for parabolic equations.\S 1.An abstract formulation of the monotone case.Let $\beta$ be a maximal monotone graph in $R\times R$ which contains the origin.If the pair $(s, t)\in\beta$ , we write $t\in\beta(s)$ .Let $\Omega$ be any measure space.We denote by $\Vert\Vert_{p}$ the norm in $L^{p}(\Omega)$ .Let $A$ be an unbounded linear operator on $L^{1}(\Omega)$ which satisfies the following conditions. (I) It is a (closed) operator with dense domain $D(A)$ in $L^{1}(\Omega)$ ; for any(II) For any $\lambda>0$ and $f\in L^{1}(\Omega)$ , $\sup_{\Omega}(I+\lambda A)^{-1}f\leqq\max\{0, \sup_{\Omega}f\}$ .(By " sup" we mean the essential supremum.If $supf=\infty$ , assumption (II) is empty.) (III) There exists $\alpha>0$ such that $\alpha\Vert u\Vert_{1}\leqq\Vert$ Au $\Vert_{1}$ for all $u\in D(A)$ .THEOREM 1.For every $f\in L^{1}(\Omega)$ , there exists a unique $u\in D(A)$ such that (2) Au $(x)+\beta(u(x))\ni f(x)$ $a$ .$e$ .Moreover, if $f,$ $f\in L^{1}(\Omega)$ and $u,$ \^u are the correspOnding solutions of (2), then(3)In particular, (4)LEMMA 2. Let $\gamma$ be a maximal monotone graph in $R\times R$ which contains the origin.Assume that $A$ satisfies (I) and (II).Let $ 1\leqq P\leqq\infty$ and $P^{\prime}=p/(p-1)$ , $ p^{\prime}=\infty$ if $p=1$ .Let $u\in D(A)\cap L^{p}(\Omega)$ with $Au\in L^{p}(\Omega)$ .Let $g\in L^{p\prime}(\Omega)$ be such that $g(x)\in\gamma(u(x))a$ .$e$ .Then $\int_{\Omega}Au(x)g(x)dx\geqq 0$ .PROOF OF THEOREM 1.We denote, for $u$ and $f\in L^{1}(\Omega),$ $f\in Bu$ whenever $f(x)\in\beta(u(x))a$ .$e$ .We first establish (3) which implies (4) and the uniqueness.Let $g=f-Au\in Bu$ and $\hat{g}=\hat{f}-A\hat{u}\in B\text{{\it \^{u}}}$ .We multiply the equation $*(I)$ is equivalent to $-A$ generating a linear contraction semi.group in $L^{1}(\Omega)$ .

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BIow-up solutions of some nonlinear elliptic problems

1997-01-01

Juan Luís Vázquez , Haïm Brézis

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Uniform estimates and blow–up behavior for solutions of −δ(u)=v(x)e<sup>u</sup>in two dimensions

1991-01-01

Haïm Brézis , Frank Merle

Click to increase image sizeClick to decrease image size Additional informationNotes on contributors Click to increase image sizeClick to decrease image size Additional informationNotes on contributors

A note on limiting cases of sobolev embeddings and convolution inequalities

1980-01-01

Haïm Brézis , Stephen Wainger

(1980). A note on limiting cases of sobolev embeddings and convolution inequalities. Communications in Partial Differential Equations: Vol. 5, No. 7, pp. 773-789. (1980). A note on limiting cases of sobolev embeddings and convolution inequalities. Communications in Partial Differential Equations: Vol. 5, No. 7, pp. 773-789.

Remarks on sublinear elliptic equations

1986-01-01

Haïm Brézis , Luc Oswald

Nonlinear Schrödinger evolution equations

1980-01-01

Haïm Brézis , Thierry Gallouët

Monotonicity Methods in Hilbert Spaces and Some Applications to Nonlinear Partial Differential Equations

1971-01-01

Haïm Brézis

Remarks on finding critical points

1991-10-01

Haïm Brézis , Louis Nirenberg

Another look at Sobolev spaces

2001-01-01

Jean Bourgain , Haïm Brézis , Petru Mironescu

The standard seminorm in the space $W^{s,p}$, with $s$<$1$, does not converge, when $s$ approaches $1$, to the corresponding $W^{1,p}$ seminorm. We prove that continuity is restored provided we multiply … The standard seminorm in the space $W^{s,p}$, with $s$<$1$, does not converge, when $s$ approaches $1$, to the corresponding $W^{1,p}$ seminorm. We prove that continuity is restored provided we multiply the $W^{s,p}$ seminorm by an appropriate factor.

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Remarks on the Euler equation

1974-04-01

J P Bourguignon , Haïm Brézis

Sublinear elliptic equations in ℝn

1992-12-01

Haïm Brézis , S. Kamin

Asymptotics for the minimization of a Ginzburg-Landau functional

1993-05-01

Fabrice Béthuel , Haïm Brézis , Fr�d�ric H�lein

Sur la régularité de la solution d'inéquations elliptiques

1968-01-01

Haïm Brézis , Guido Stampacchia

programme Numérisation de documents anciens mathématiques http://www.numdam.org/l54 H. BREZIS ET G programme Numérisation de documents anciens mathématiques http://www.numdam.org/l54 H. BREZIS ET G

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Hardy's inequalities revisited

1997-01-01

Haïm Brézis , Moshe Marcus

A nonlinear heat equation with singular initial data

1996-12-01

Haïm Brézis , Thierry Cazenave

Degree theory and BMO; part II: Compact manifolds with boundaries

1996-03-01

Haïm Brézis , Louis Nirenberg

H1 versus C1 local minimizers

1993-01-01

Haïm Brézis , Louis Nirenberg

Produits infinis de resolvantes

1978-12-01

Haïm Brézis , Pierre‐Louis Lions

Degree theory and BMO; part I: Compact manifolds without boundaries

1995-09-01

Haïm Brézis , Louis Nirenberg

Perturbations of nonlinear maximal monotone sets in banach space

1970-01-01

Haïm Brézis , Michael G. Crandall , A. Pazy

Convergence and approximation of semigroups of nonlinear operators in Banach spaces

1972-01-01

Haïm Brézis , A. Pazy

On a class of superlinear elliptic problems

1977-01-01

Haïm Brézis , R. E. Turner

(1977). On a class of superlinear elliptic problems. Communications in Partial Differential Equations: Vol. 2, No. 6, pp. 601-614. (1977). On a class of superlinear elliptic problems. Communications in Partial Differential Equations: Vol. 2, No. 6, pp. 601-614.

Removable singularities for some nonlinear elliptic equations

1980-03-01

Haïm Brézis , Лаурент Верон

Multiple solutions of <i>H</i>‐systems and Rellich's conjecture

1984-03-01

Haïm Brézis , Jean‐Michel Coron

Semilinear equations in ? N without condition at infinity

1984-10-01

Haïm Brézis

Some simple nonlinear PDE's without solutions

1998-06-01

Haïm Brézis , Xavier Cabré

Blow up for $u_t-\Delta u=g(u)$ revisited

1996-01-01

Haïm Brézis , Thierry Cazenave , Yvan Martel +1 more

A general principle on ordered sets in nonlinear functional analysis

1976-09-01

Haïm Brézis , Felix E. Browder

Limiting embedding theorems forW s,p whens ↑ 1 and applications

2002-12-01

Jean Bourgain , Haïm Brézis , Petru Mironescu

Gagliardo-Nirenberg, composition and products in fractional Sobolev spaces

2001-12-01

Haïm Brézis , Petru Mironescu

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Periodic solutions of nonlinear vibrating strings and duality principles

1983-01-01

Haïm Brézis

Introduction.Our lecture deals with the study of T-periodic solutions for the nonlinear vibrating string equation:Here g denotes a continuous function on R such that g(0) = 0 and f(x, t) … Introduction.Our lecture deals with the study of T-periodic solutions for the nonlinear vibrating string equation:Here g denotes a continuous function on R such that g(0) = 0 and f(x, t) is a given T-periodic function of t.Problem (1) may be viewed as an infinite-dimensional Hamiltonian system (let us recall that H. Poincaré has abundantly investigated the question of periodic solutions for finite-dimensional Hamiltonian systems; see [50]).Indeed if we set p = u and q -u n then (1) becomes è(;H-";Hï)where the Hamiltonian H is defined on the space #0(0, w) X L 2 (0, IT) byand G denotes a primitive of g.We shall be concerned with two distinct questions.Question 1. Existence of forced vibrations; that is, given ƒ(JC, /) find at least one solution of (1).Question 2. Existence of free vibrations (or "breathers"); that is, assume ƒ = 0 and find at least one nonzero solution of (1).

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On a characterization of flow‐invariant sets

1970-03-01

Haïm Brézis

Lifting in Sobolev spaces

2000-12-01

Jean Bourgain , Haïm Brézis , Petru Mironescu

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A remark on Ky fan's minimax principle

2008-01-01

Haïm Brézis , Louis Nirenberg , G. Stampacchia

Forced vibrations for a nonlinear wave equation

1978-01-01

Haïm Brézis , Louis Nirenberg

Image d’une somme d’operateurs monotones et applications

1976-02-01

Haïm Brézis , Alain Haraux

On the equation 𝑑𝑖𝑣𝑌=𝑓 and application to control of phases

2002-11-26

Jean Bourgain , Haïm Brézis

The main result is the following. Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega"> <mml:semantics> <mml:mi mathvariant="normal">Ω</mml:mi> <mml:annotation encoding="application/x-tex">\Omega</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a bounded Lipschitz domain in … The main result is the following. Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega"> <mml:semantics> <mml:mi mathvariant="normal">Ω</mml:mi> <mml:annotation encoding="application/x-tex">\Omega</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a bounded Lipschitz domain in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R Superscript d"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>d</mml:mi> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb {R}^{d}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d greater-than-or-equal-to 2"> <mml:semantics> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">d\geq 2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Then for every <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f element-of upper L Superscript d Baseline left-parenthesis normal upper Omega right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>∈</mml:mo> <mml:msup> <mml:mi>L</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>d</mml:mi> </mml:mrow> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="normal">Ω</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">f\in L^{d}(\Omega )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral f equals 0"> <mml:semantics> <mml:mrow> <mml:mo>∫</mml:mo> <mml:mi>f</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\int f =0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, there exists a solution <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u element-of upper C Superscript 0 Baseline left-parenthesis normal upper Omega overbar right-parenthesis intersection upper W Superscript 1 comma d Baseline left-parenthesis normal upper Omega right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>u</mml:mi> <mml:mo>∈</mml:mo> <mml:msup> <mml:mi>C</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>0</mml:mn> </mml:mrow> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi mathvariant="normal">Ω</mml:mi> <mml:mo stretchy="false">¯</mml:mo> </mml:mover> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> <mml:mo>∩</mml:mo> <mml:msup> <mml:mi>W</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mi>d</mml:mi> </mml:mrow> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="normal">Ω</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">u\in C^{0}(\bar \Omega )\cap W^{1, d}(\Omega )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of the equation div <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u equals f"> <mml:semantics> <mml:mrow> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mi>f</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">u=f</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega"> <mml:semantics> <mml:mi mathvariant="normal">Ω</mml:mi> <mml:annotation encoding="application/x-tex">\Omega</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, satisfying in addition <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u equals 0"> <mml:semantics> <mml:mrow> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">u=0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="partial-differential normal upper Omega"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="normal">∂</mml:mi> <mml:mi mathvariant="normal">Ω</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\partial \Omega</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and the estimate <disp-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar u double-vertical-bar Subscript upper L Sub Superscript normal infinity Baseline plus double-vertical-bar u double-vertical-bar Subscript upper W Sub Superscript 1 comma d Baseline less-than-or-equal-to upper C double-vertical-bar f double-vertical-bar Subscript upper L Sub Superscript d"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false">‖</mml:mo> <mml:mi>u</mml:mi> <mml:msub> <mml:mo fence="false" stretchy="false">‖</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>L</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">∞</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> </mml:msub> <mml:mo>+</mml:mo> <mml:mo fence="false" stretchy="false">‖</mml:mo> <mml:mi>u</mml:mi> <mml:msub> <mml:mo fence="false" stretchy="false">‖</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>W</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mi>d</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> </mml:msub> <mml:mo>≤</mml:mo> <mml:mi>C</mml:mi> <mml:mo fence="false" stretchy="false">‖</mml:mo> <mml:mi>f</mml:mi> <mml:msub> <mml:mo fence="false" stretchy="false">‖</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>L</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>d</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">\begin{equation*}\Vert u\Vert _{L^{\infty }}+\Vert u\Vert _{W^{1, d}}\leq C\Vert f\Vert _{L^{d}} \end{equation*}</mml:annotation> </mml:semantics> </mml:math> </disp-formula> where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C"> <mml:semantics> <mml:mi>C</mml:mi> <mml:annotation encoding="application/x-tex">C</mml:annotation> </mml:semantics> </mml:math> </inline-formula> depends only on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega"> <mml:semantics> <mml:mi mathvariant="normal">Ω</mml:mi> <mml:annotation encoding="application/x-tex">\Omega</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. However one cannot choose <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u"> <mml:semantics> <mml:mi>u</mml:mi> <mml:annotation encoding="application/x-tex">u</mml:annotation> </mml:semantics> </mml:math> </inline-formula> depending linearly on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f"> <mml:semantics> <mml:mi>f</mml:mi> <mml:annotation encoding="application/x-tex">f</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Our proof is constructive, but nonlinear—which is quite surprising for such an elementary linear PDE. When <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d equals 2"> <mml:semantics> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">d=2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> there is a simpler proof by duality—hence nonconstructive.

Minimum action solutions of some vector field equations

1984-03-01

Haïm Brézis , Élliott H. Lieb

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Gagliardo–Nirenberg inequalities and non-inequalities: The full story

2017-11-24

Petru Mironescu , Haïm Brézis

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Sobolev Inequalities with Remainder Terms

2002-01-01

Haïm Brézis , Élliott H. Lieb

Elliptic equations with limiting sobolev exponents—the impact of topology

1986-01-01

Haïm Brézis

Sobolev inequalities with remainder terms

1985-06-01

Haïm Brézis , Élliott H. Lieb

New estimates for elliptic equations and Hodge type systems

2007-06-30

Jean Bourgain , Haïm Brézis

We establish new estimates for the Laplacian, the div-curl system, and more general Hodge systems in arbitrary dimension n , with data in L1 . We also present related results … We establish new estimates for the Laplacian, the div-curl system, and more general Hodge systems in arbitrary dimension n , with data in L1 . We also present related results concerning differential forms with coefficients in the limiting Sobolev space W1,n .

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Families of functionals representing Sobolev norms

2024-04-24

Haïm Brézis , Andreas Seeger , Jean Van Schaftingen +1 more

We obtain new characterizations of the Sobolev spaces Ẇ 1, p ‫ޒ(‬ N ) and the bounded variation space ḂV(‫ޒ‬ N ).The characterizations are in terms of the functionals ν … We obtain new characterizations of the Sobolev spaces Ẇ 1, p ‫ޒ(‬ N ) and the bounded variation space ḂV(‫ޒ‬ N ).The characterizations are in terms of the functionals ν γ (E λ,γ / p [u]), whereand the measure ν γ is given by dν γ (x, y) = |x -y| γ -N dx dy.We provide characterizations which involve the L p,∞ -quasinorms sup λ>0 λν γ (E λ,γ / p [u]) 1/ p and also exact formulas via corresponding limit functionals, with the limit for λ → ∞ when γ > 0 and the limit for λ → 0 + when γ < 0. The results unify and substantially extend previous work by Nguyen and by Brezis, Van Schaftingen and Yung.For p > 1 the characterizations hold for all γ ̸ = 0.For p = 1 the upper bounds for the L 1,∞ quasinorms fail in the range γ ∈ [-1, 0); moreover, in this case the limit functionals represent the L 1 norm of the gradient for C ∞ c -functions but not for generic Ẇ 1,1 -functions.For this situation we provide new counterexamples which are built on self-similar sets of dimension γ + 1.For γ = 0 the characterizations of Sobolev spaces fail; however, we obtain a new formula for the Lipschitz norm via the expressions ν 0 (E λ,0 [u]).

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Non-local approximations of the gradient

2024-04-08

Haïm Brézis , Petru Mironescu

We revisit the proofs of a few basic results concerning non-local approximations of the gradient. A typical such result asserts that, if (ρ ε ) is a radial approximation to … We revisit the proofs of a few basic results concerning non-local approximations of the gradient. A typical such result asserts that, if (ρ ε ) is a radial approximation to the identity in ℝ N and u belongs to a homogeneous Sobolev space W ˙ 1,p , then

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A brief history of the Jacobian

2023-02-20

Haïm Brézis , Jean Mawhin , Petru Mironescu

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A brief history of the Jacobian

2023-01-13

Haïm Brézis , Jean Mawhin , Petru Mironescu

In his pioneering work, Jacobi discovered two remarkable identities related to the Jacobian. The first one asserts that the Jacobian has a divergence structure. The second one, that some vector … In his pioneering work, Jacobi discovered two remarkable identities related to the Jacobian. The first one asserts that the Jacobian has a divergence structure. The second one, that some vector fields involving the cofactors of the Jacobian are divergence free. We illustrate the fundamental impact of these properties on research, from the times of Jacobi to our days.

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Sobolev spaces revisited

2022-08-30

Haïm Brézis , Andreas Seeger , Jean Van Schaftingen +1 more

We describe a recent, one-parameter family of characterizations of Sobolev and BV functions on \mathbb{R}^n n, using sizes of superlevel sets of suitable difference quotients. This provides an alternative point … We describe a recent, one-parameter family of characterizations of Sobolev and BV functions on \mathbb{R}^n n, using sizes of superlevel sets of suitable difference quotients. This provides an alternative point of view to the BBM formula by Bourgain, Brezis, and Mironescu, and complements in the case of BV some results of Cohen, Dahmen, Daubechies, and DeVore about the sizes of wavelet coefficients of such functions. An application towards Gagliardo–Nirenberg interpolation inequalities is then given. We also establish a related one-parameter family of formulae for the L^p norm of functions in L^p(\mathbb{R}^n) .

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Sobolev spaces revisited

2022-01-01

Haïm Brézis , Andreas Seeger , Jean Van Schaftingen +1 more

We describe a recent, one-parameter family of characterizations of Sobolev and BV functions on $\mathbb{R}^n$, using sizes of superlevel sets of suitable difference quotients. This provides an alternative point of … We describe a recent, one-parameter family of characterizations of Sobolev and BV functions on $\mathbb{R}^n$, using sizes of superlevel sets of suitable difference quotients. This provides an alternative point of view to the BBM formula by Bourgain, Brezis and Mironescu, and complements in the case of BV some results of Cohen, Dahmen, Daubechies and DeVore about the sizes of wavelet coefficients of such functions. An application towards Gagliardo-Nirenberg interpolation inequalities is then given. We also establish a related one-parameter family of formulae for the $L^p$ norm of functions in $L^p(\mathbb{R}^n)$.

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A sharp relative isoperimetric inequality for the square

2021-11-25

Haïm Brézis , Alfred M. Bruckstein⋆

We compute the exact value of the least “relative perimeter” of a shape S, with a given area, contained in a unit square; the relative perimeter of S being the … We compute the exact value of the least “relative perimeter” of a shape S, with a given area, contained in a unit square; the relative perimeter of S being the length of the boundary of S that does not touch the border of the square.

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Going to Lorentz when fractional Sobolev, Gagliardo and Nirenberg estimates fail

2021-06-29

Haïm Brézis , Jean Van Schaftingen , Po‐Lam Yung

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A surprising formula for Sobolev norms

2021-02-15

Haïm Brézis , Jean Van Schaftingen , Po‐Lam Yung

Significance The Sobolev spaces, introduced in the 1930s, have become ubiquitous in analysis and applied mathematics. They involve <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:msup><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msup></mml:math> norms of the gradient of a function <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" … Significance The Sobolev spaces, introduced in the 1930s, have become ubiquitous in analysis and applied mathematics. They involve <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:msup><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msup></mml:math> norms of the gradient of a function <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mi>u</mml:mi></mml:math> . We present an alternative point of view where derivatives are replaced by appropriate finite differences and the Lebesgue space <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:msup><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msup></mml:math> is replaced by the slightly larger Marcinkiewicz space <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:msup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msup></mml:math> (aka weak <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:msup><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msup></mml:math> space)—a popular tool in harmonic analysis. Surprisingly, these spaces coincide with the standard Sobolev spaces, a fact which sheds additional light onto these classical objects and should have numerous applications. In particular, it rectifies some well-known irregularities occurring in the theory of fractional Sobolev spaces. The proof relies on original calculus inequalities which might be useful in other situations.

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Families of functionals representing Sobolev norms

2021-01-01

Haïm Brézis , Andreas Seeger , Jean Van Schaftingen +1 more

We obtain new characterizations of the Sobolev spaces $\dot W^{1,p}(\mathbb{R}^N)$ and the bounded variation space $\dot{BV}(\mathbb{R}^N)$. The characterizations are in terms of the functionals $\nu_{\gamma} (E_{\lambda,\gamma/p}[u])$ where \[ E_{\lambda,\gamma/p}[u]= \Big\{(x,y … We obtain new characterizations of the Sobolev spaces $\dot W^{1,p}(\mathbb{R}^N)$ and the bounded variation space $\dot{BV}(\mathbb{R}^N)$. The characterizations are in terms of the functionals $\nu_{\gamma} (E_{\lambda,\gamma/p}[u])$ where \[ E_{\lambda,\gamma/p}[u]= \Big\{(x,y )\in \mathbb{R}^N \times \mathbb{R}^N \colon x \neq y, \, \frac{|u(x)-u(y)|}{|x-y|^{1+\gamma/p}}>\lambda\Big\} \] and the measure $\nu_{\gamma}$ is given by $\mathrm{d} \nu_\gamma(x,y)=|x-y|^{\gamma-N} \mathrm{d} x \mathrm{d} y$. We provide characterizations which involve the $L^{p,\infty}$-quasi-norms $\sup_{\lambda>0} \lambda \, \nu_{\gamma} (E_{\lambda,\gamma/p}[u]) ^{1/p}$ and also exact formulas via corresponding limit functionals, with the limit for $\lambda\to\infty$ when $\gamma>0$ and the limit for $\lambda\to 0^+$ when $\gamma<0$. The results unify and substantially extend previous work by Nguyen and by Brezis, Van Schaftingen and Yung. For $p>1$ the characterizations hold for all $\gamma \neq 0$. For $p=1$ the upper bounds for the $L^{1,\infty}$ quasi-norms fail in the range $\gamma\in [-1,0) $; moreover in this case the limit functionals represent the $L^1$ norm of the gradient for $C^\infty_c$-functions but not for generic $\dot W^{1,1}$-functions. For this situation we provide new counterexamples which are built on self-similar sets of dimension $\gamma+1$. For $\gamma=0$ the characterizations of Sobolev spaces fail; however we obtain a new formula for the Lipschitz norm via the expressions $\nu_0(E_{\lambda,0}[u])$.

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Density

2021-01-01

Haïm Brézis , Petru Mironescu

We investigate here density questions. For real-valued Sobolev spaces, $$C^\infty (\overline{\varOmega }; \mathbb R)$$ is dense in $$W^{s,p}(\varOmega ; \mathbb R)$$ , for any $$s>0$$ and $$1\le p<\infty $$ . … We investigate here density questions. For real-valued Sobolev spaces, $$C^\infty (\overline{\varOmega }; \mathbb R)$$ is dense in $$W^{s,p}(\varOmega ; \mathbb R)$$ , for any $$s>0$$ and $$1\le p<\infty $$ . This need not be true for the Sobolev spaces $$W^{s,p}(\varOmega ; {\mathscr {N}})$$ , where $$\mathscr {N}$$ is a manifold. In particular, this is not always the case when $${\mathscr {N}}={\mathbb S}^1$$ . We present the optimal conditions on s and p so that $$C^{\infty }(\overline{\varOmega }; {\mathbb S}^1)$$ is dense in $$W^{s,p}(\varOmega ; {\mathbb S}^1)$$ .

Domains with topology

2021-01-01

Haïm Brézis , Petru Mironescu

We investigate the impact of the topology of $$\varOmega $$ on the main topics presented above (existence of lifting, relaxed energy, density of smooth maps, etc.). We investigate the impact of the topology of $$\varOmega $$ on the main topics presented above (existence of lifting, relaxed energy, density of smooth maps, etc.).

Lifting in fractional Sobolev spaces and in $$\textit{VMO}\, $$

2021-01-01

Haïm Brézis , Petru Mironescu

This chapter is a follow-up to Chapter 1. We consider again maps $u:\varOmega \rightarrow {\mathbb S}^1$, with $\varOmega \subset \mathbb R^N$, $N\ge 1$, smooth bounded simply connected, and our goal … This chapter is a follow-up to Chapter 1. We consider again maps $u:\varOmega \rightarrow {\mathbb S}^1$, with $\varOmega \subset \mathbb R^N$, $N\ge 1$, smooth bounded simply connected, and our goal is to find a lifting $\varphi $ "as smooth as u permits." The difference with Chapter 1 is that u belongs either to a fractional Sobolev space $W^{s,p}$ (whose definition is recalled in Section 5.1), or to the space $\textit{VMO}\, $ (functions of vanishing mean oscillation, whose definition is recalled in Section 5.2.1 below).

Lifting in $$W^{1,p}$$

2021-01-01

Haïm Brézis , Petru Mironescu

We investigate here the question of lifting within the framework of Sobolev spaces $W^{1,p}(\varOmega ; {\mathbb S}^1)$, where $\varOmega \subset \mathbb R^N$ is, say, a ball. In the process, we … We investigate here the question of lifting within the framework of Sobolev spaces $W^{1,p}(\varOmega ; {\mathbb S}^1)$, where $\varOmega \subset \mathbb R^N$ is, say, a ball. In the process, we are led to the introduction of two fundamental tools, the distributional Jacobian $Ju=1/2\, d(u\wedge du)$, and its Wasserstein norm $\varSigma (u)$, which are ubiquitous throughout the book.

The geometry of Ju and $$\varSigma (u)$$ in 3D (and higher); line singularities and minimal surfaces

2021-01-01

Haïm Brézis , Petru Mironescu

This chapter is the 3D (and higher) counterpart of the previous one. Here, geometry plays a crucial role. The (singular) points $a_i$ are replaced by curves, connections are replaced by … This chapter is the 3D (and higher) counterpart of the previous one. Here, geometry plays a crucial role. The (singular) points $a_i$ are replaced by curves, connections are replaced by surfaces with prescribed boundaries or more general geometric objects (such as 2-rectifiable currents), minimal connections are replaced by area-minimizing surfaces spanned by curves (as in the classical Plateau problem).

Appendices

2021-01-01

Haïm Brézis , Petru Mironescu

We recall standard properties of Sobolev spaces used throughout the book, and we gather the most technical parts of some of the proofs presented in the main text. We recall standard properties of Sobolev spaces used throughout the book, and we gather the most technical parts of some of the proofs presented in the main text.

Factorization

2021-01-01

Haïm Brézis , Petru Mironescu

In Chapter 1, we saw that a map in $W^{1, 1}(\varOmega ; {\mathbb S}^1)$, where $N\ge 2$, need not have a phase in $W^{1,1}$. Recall that a typical obstruction arises … In Chapter 1, we saw that a map in $W^{1, 1}(\varOmega ; {\mathbb S}^1)$, where $N\ge 2$, need not have a phase in $W^{1,1}$. Recall that a typical obstruction arises when $N=2$ and $0\in \varOmega $, with $v_0(x):=x/|x|$. We know that.

Traces

2021-01-01

Haïm Brézis , Petru Mironescu

We present a complete trace theory for $${\mathbb S}^1$$ -valued maps. When $$s>0$$ is not an integer and $$1\le p<\infty $$ , standard trace theory for real-valued maps asserts that … We present a complete trace theory for $${\mathbb S}^1$$ -valued maps. When $$s>0$$ is not an integer and $$1\le p<\infty $$ , standard trace theory for real-valued maps asserts that $$\begin{aligned}{\text {tr }}_{\varOmega } W^{s+1/p,p}(\varOmega \times (0,1) ; \mathbb R)=W^{s,p}(\varOmega ; \mathbb R), \end{aligned}$$ where we identify $$\varOmega $$ with $$\varOmega \times \{ 0\}$$ . When $$\mathbb R$$ is replaced by a manifold $$\mathscr {N}$$ , in general we only have the inclusion $$\begin{aligned}{\text {tr }}_{\varOmega } W^{s+1/p,p}(\varOmega \times (0,1) ; {\mathscr {N}})\subset W^{s,p}(\varOmega ; {\mathscr {N}}), \end{aligned}$$ and equality may fail.

Dirichlet problems. Gaps. Infinite energies

2021-01-01

Haïm Brézis , Petru Mironescu

We investigate minimization problems of the form $$\begin{aligned} \min \left\{ \int _\varOmega |\nabla u|^p;\, u\in W^{1,p}_g(\varOmega ; {\mathbb S}^1)\right\} , \end{aligned}$$where $1\le p<\infty $ and $g\in C^\infty (\partial \varOmega ; … We investigate minimization problems of the form $$\begin{aligned} \min \left\{ \int _\varOmega |\nabla u|^p;\, u\in W^{1,p}_g(\varOmega ; {\mathbb S}^1)\right\} , \end{aligned}$$where \(1\le p<\infty $ and $g\in C^\infty (\partial \varOmega ; {\mathbb S}^1)$ is a given boundary condition (satisfying also $\deg g=0$ when $N=2$).

The geometry of Ju and $$\varSigma (u)$$ in 2D; point singularities and minimal connections

2021-01-01

Haïm Brézis , Petru Mironescu

In this chapter, we give illuminating geometric interpretations for Ju and $\varSigma (u)$ when $N=2$. In 2D, we naturally identify the 2-form Ju with a scalar distribution. Assuming for example … In this chapter, we give illuminating geometric interpretations for Ju and $\varSigma (u)$ when $N=2$. In 2D, we naturally identify the 2-form Ju with a scalar distribution. Assuming for example that $u\in W^{1,1}(\varOmega ; {\mathbb S}^1)$ is smooth except at some point $a\in \varOmega $, and that the winding number, $\deg (u, a)$, of u on small circles around a, equals one, we have $$\begin{aligned} Ju=\pi \delta _a \ \text {in }{\mathscr {D}}'(\varOmega ). \end{aligned}$$More generally, if $u\in W^{1,1}(\varOmega ; {\mathbb S}^1)$ is a "nice" map, i.e., continuous on $\varOmega $ except at a finite number of distinct points $a_1,\ldots , a_k$, we have $$\begin{aligned} Ju=\pi \sum _{j=1}^k \deg (u, a_j)\, \delta _{a_j}\ \text {in }{\mathscr {D}}'(\varOmega ). \end{aligned}$$Next, we turn to the geometric interpretation of $\varSigma (u)$ as a "minimal length" required to "connect the singularities."

A digression: sphere-valued maps

2021-01-01

Haïm Brézis , Petru Mironescu

The purpose of this chapter is to explain rather informally how some of the tools and results presented in the previous chapters extend to maps $u:\varOmega \rightarrow {\mathbb S}^k$, where … The purpose of this chapter is to explain rather informally how some of the tools and results presented in the previous chapters extend to maps $u:\varOmega \rightarrow {\mathbb S}^k$, where $\varOmega \subset \mathbb {R}^N$ is a smooth bounded open set, $N\ge 2$ and $1\le k\le N-1$. To avoid any complication, we assume throughout this chapter that $\varOmega $ is diffeomorphic to a ball.

Uniqueness of lifting and beyond

2021-01-01

Haïm Brézis , Petru Mironescu

So far, wehave been concerned with the existence of a lifting $\varphi :\varOmega \rightarrow \mathbb R$ for a given $u:\varOmega \rightarrow {\mathbb S}^1$. A natural question is whether such $\varphi … So far, wehave been concerned with the existence of a lifting \(\varphi :\varOmega \rightarrow \mathbb R$ for a given $u:\varOmega \rightarrow {\mathbb S}^1$. A natural question is whether such $\varphi $ is unique (mod $2\pi $). More precisely, assume that we have two liftings, $\varphi _1$, $\varphi _2$. Then, $\varphi _1(x)-\varphi _2(x)=2\pi k(x)$ for some $k:\varOmega \rightarrow \mathbb Z$. Therefore, we are led to the question of finding minimal assumptions on a measurable function $k:\varOmega \rightarrow \mathbb Z$ implying that k must be constant. Clearly, continuity is sufficient, but, as we are going to see, constancy holds for a surprisingly large class of functions.

Applications of the factorization

2021-01-01

Haïm Brézis , Petru Mironescu

We present various applications of the factorization. Arguably the most spectacular one is the possibility of giving a "robust" definition to $u\wedge d u$ when $u\in W^{1/p,p}(\varOmega ; {\mathbb S}^1)$, … We present various applications of the factorization. Arguably the most spectacular one is the possibility of giving a "robust" definition to $u\wedge d u$ when $u\in W^{1/p,p}(\varOmega ; {\mathbb S}^1)$, with $1<p<\infty $. As a byproduct, we also obtain a robust definition of the distribution Ju for such u.

Degree

2021-01-01

Haïm Brézis , Petru Mironescu

We revisit the notion of topological degree $\deg f$ (aka index or winding number) for maps $f:{\mathbb S}^1\rightarrow {\mathbb S}^1$. This is a classical concept when f is continuous: $\deg … We revisit the notion of topological degree \(\deg f$ (aka index or winding number) for maps $f:{\mathbb S}^1\rightarrow {\mathbb S}^1$. This is a classical concept when f is continuous: $\deg f$ counts "how many times $f({\mathbb S}^1)$ covers ${\mathbb S}^1$, taking into account algebraic multiplicity." One can still give a robust definition for $\deg f$ when f belongs merely to $\text { VMO}\, ({\mathbb S}^1; {\mathbb S}^1)$, and thus, by the Sobolev embeddings, for maps in the critical spaces $W^{1/p,p}({\mathbb S}^1; {\mathbb S}^1)$, with $1<p<\infty $. We establish some basic properties of this degree.

A surprising formula for Sobolev norms and related topics

2020-03-11

Haïm Brézis , Jean Van Schaftingen , Po‐Lam Yung

We establish the equivalence between the Sobolev semi-norm $\|\nabla u\|_{L^p}$ and a quantity obtained when replacing the strong $L^p$ by a weak $L^p$ norm in the Gagliardo semi-norm $|u|_{W^{s,p}}$ computed … We establish the equivalence between the Sobolev semi-norm $\|\nabla u\|_{L^p}$ and a quantity obtained when replacing the strong $L^p$ by a weak $L^p$ norm in the Gagliardo semi-norm $|u|_{W^{s,p}}$ computed at $s = 1$. As corollaries we derive alternative estimates in some exceptional cases (involving $W^{1,1}$) where the anticipated fractional Sobolev and Gagliardo-Nirenberg inequalities fail.

Γ-convergence of non-local, non-convex functionals in one dimension

2019-10-11

Haïm Brézis , Hoài-Minh Nguyên

We study the [Formula: see text]-convergence of a family of non-local, non-convex functionals in [Formula: see text] for [Formula: see text], where [Formula: see text] is an open interval. We … We study the [Formula: see text]-convergence of a family of non-local, non-convex functionals in [Formula: see text] for [Formula: see text], where [Formula: see text] is an open interval. We show that the limit is a multiple of the [Formula: see text] semi-norm to the power [Formula: see text] when [Formula: see text] (respectively, the [Formula: see text] semi-norm when [Formula: see text]). In dimension one, this extends earlier results which required a monotonicity condition.

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Non-local, non-convex functionals converging to Sobolev norms

2019-09-24

Haïm Brézis , Hoài-Minh Nguyên

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$\Gamma$-convergence of non-local, non-convex functionals in one dimension

2019-09-05

Haïm Brézis , Hoài-Minh Nguyên

We study the $\Gamma$-convergence of a family of non-local, non-convex functionals in $L^p(I)$ for $p \ge 1$, where $I$ is an open interval. We show that the limit is a … We study the $\Gamma$-convergence of a family of non-local, non-convex functionals in $L^p(I)$ for $p \ge 1$, where $I$ is an open interval. We show that the limit is a multiple of the $W^{1, p}(I)$ semi-norm to the power $p$ when $p>1$ (resp. the $BV(I)$ semi-norm when $p=1$). In dimension one, this extends earlier results which required a monotonicity condition.

Non-local, non-convex functionals converging to Sobolev norms

2019-09-05

Haïm Brézis , Hoài-Minh Nguyên

We study the pointwise convergence and the $\Gamma$-convergence of a family of non-local, non-convex functionals $\Lambda_\delta$ in $L^p(\Omega)$ for $p>1$. We show that the limits are multiples of $\int_{\Omega} |\nabla … We study the pointwise convergence and the $\Gamma$-convergence of a family of non-local, non-convex functionals $\Lambda_\delta$ in $L^p(\Omega)$ for $p>1$. We show that the limits are multiples of $\int_{\Omega} |\nabla u|^p$. This is a continuation of our previous work where the case $p=1$ was considered.

The Plateau problem from the perspective of optimal transport

2019-07-01

Haïm Brézis , Petru Mironescu

Both optimal transport and minimal surfaces have received much attention in recent years. We show that the methodology introduced by Kantorovich on the Monge problem can, surprisingly, be adapted to … Both optimal transport and minimal surfaces have received much attention in recent years. We show that the methodology introduced by Kantorovich on the Monge problem can, surprisingly, be adapted to questions involving least area, e.g., Plateau-type problems as investigated by Federer. Le transport optimal, ainsi que les surfaces minimales, ont été abondamment étudiés au cours de ces dernières décennies. Nous mettons en évidence une analogie surprenante, au niveau méthodologique, entre l'approche de Kantorovich pour le problème de Monge et la minimisation de l'aire dans des problèmes géométriques de type Plateau étudiés par Federer.

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Where Sobolev interacts with Gagliardo–Nirenberg

2019-03-12

Haïm Brézis , Petru Mironescu

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Remarks on some minimization problems associated with BV norms

2019-01-01

Haïm Brézis

The purpose of this paper is twofold. Firstly I present an optimal regularity result for minimizers of a $ 1D $ convex functional involving the BV-norm, under Neumann boundary condition. … The purpose of this paper is twofold. Firstly I present an optimal regularity result for minimizers of a $ 1D $ convex functional involving the BV-norm, under Neumann boundary condition. This functional is a simplified version of models occuring in Image Processing. Secondly I investigate the existence of minimizers for the same functional under Dirichlet boundary condition. Surprisingly, this turns out to be a delicate issue, which is still widely open.

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Existence of solutions for a nonhomogeneous semilinear fractional laplacian problems

2019-01-01

José M. Mendoza , Haïm Brézis

$Γ$-convergence of non-local, non-convex functionals in one dimension

2019-01-01

Haïm Brézis , Hoài-Minh Nguyên

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Non-local, non-convex functionals converging to Sobolev norms

2019-01-01

Haïm Brézis , Hoài-Minh Nguyên

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Where Sobolev interacts with Gagliardo-Nirenberg

2019-01-01

Haïm Brézis , Petru Mironescu

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Minimizers of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="mml1" display="inline" overflow="scroll" altimg="si1.gif"><mml:msup><mml:mrow><mml:mi>W</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:math>-energy of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="mml2" display="inline" overflow="scroll" altimg="si2.gif"><mml:msup><mml:mrow><mml:mi mathvariant="double-struck">S</mml:mi></mml:mrow><mml:mrow…

2018-05-07

Haïm Brézis , Petru Mironescu

Radial extensions in fractional Sobolev spaces

2018-02-26

Haïm Brézis , Petru Mironescu , Itai Shafrir

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Non-local Functionals Related to the Total Variation and Connections with Image Processing

2018-01-24

Haïm Brézis , Hoài-Minh Nguyên

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Remarks on the Monge–Kantorovich problem in the discrete setting

2018-01-05

Haïm Brézis

In Optimal Transport theory, three quantities play a central role: the minimal cost of transport, originally introduced by Monge, its relaxed version introduced by Kantorovich, and a dual formulation also … In Optimal Transport theory, three quantities play a central role: the minimal cost of transport, originally introduced by Monge, its relaxed version introduced by Kantorovich, and a dual formulation also due to Kantorovich. The goal of this Note is to publicize a very elementary, self-contained argument extracted from [9], which shows that all three quantities coincide in the discrete case. En théorie du transport optimal, trois quantités jouent un rôle central : le coût minimal de transport, introduit par Monge, sa version relaxée, introduite par Kantorovich, et la formulation duale, due aussi à Kantorovich. L'objet de cette note est de mettre en avant une démonstration totalement élémentaire, extraite de [9], du fait que ces trois quantités coïncident dans le cas discret ; cette preuve ne requiert aucune connaissance préalable.

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Minimizers of the $W^{1,1}$-energy of $\mathbb S^1$-valued maps with prescribed singularities. Do they exist?

2018-01-01

Haïm Brézis , Petru Mironescu

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Distances between classes in $$W^{1,1}(\Omega ;{\mathbb {S}}^{1})$$ W 1 , 1 ( Ω ; S 1 )

2017-12-22

Haïm Brézis , Petru Mironescu , Itai Shafrir

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Gagliardo–Nirenberg inequalities and non-inequalities: The full story

2017-11-24

Petru Mironescu , Haïm Brézis

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Rigidity of optimal bases for signal spaces

2017-06-27

Haïm Brézis , David Gómez‐Castro

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Energy estimates for S-valued maps

2017-01-01

Fabrice Béthuel , Haïm Brézis , Frédéric Hélein

The lower bound for the energy of S-valued maps on Perforated domains

2017-01-01

Fabrice Béthuel , Haïm Brézis , Frédéric Hélein

Non-minimizing solutions of the Ginzburg-Landau equation

2017-01-01

Fabrice Béthuel , Haïm Brézis , Frédéric Hélein

Towards locating the singularties: bad discs and good discs

2017-01-01

Fabrice Béthuel , Haïm Brézis , Frédéric Hélein

Ginzburg-Landau Vortices

2017-01-01

Fabrice Béthuel , Haïm Brézis , Frédéric Hélein

This book is concerned with the study in two dimensions of stationary solutions of uɛ of a complex valued Ginzburg-Landau equation involving a small This book is concerned with the study in two dimensions of stationary solutions of uɛ of a complex valued Ginzburg-Landau equation involving a small

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Non-convex, non-local functionals converging to the total variation

2016-12-12

Haïm Brézis , Hoài-Minh Nguyên

We present new results concerning the approximation of the total variation, ∫Ω|∇u|, of a function u by non-local, non-convex functionals of the formΛδ(u)=∫Ω∫Ωδφ(|u(x)−u(y)|/δ)|x−y|d+1dxdy, as δ→0, where Ω is a domain … We present new results concerning the approximation of the total variation, ∫Ω|∇u|, of a function u by non-local, non-convex functionals of the formΛδ(u)=∫Ω∫Ωδφ(|u(x)−u(y)|/δ)|x−y|d+1dxdy, as δ→0, where Ω is a domain in Rd and φ:[0,+∞)→[0,+∞) is a non-decreasing function satisfying some appropriate conditions. The mode of convergence is extremely delicate, and numerous problems remain open. The original motivation of our work comes from Image Processing. Nous présentons des résultats nouveaux concernant l'approximation de la variation totale ∫Ω|∇u| d'une fonction u par des fonctionnelles non convexes et non locales de la formeΛδ(u)=∫Ω∫Ωδφ(|u(x)−u(y)|/δ)|x−y|d+1dxdy, quand δ→0, où Ω est un domaine de Rd et φ:[0,+∞)→[0,+∞) est une fonction croissante vérifiant certaines hypothèses. Le mode de convergence est extrêmement délicat et de nombreux problèmes restent ouverts. La motivation provient du traitement d'images.

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Coauthor	Papers Together
Petru Mironescu	56
Augusto C. Ponce	23
Jean Bourgain	20
Fabrice Béthuel	19
Hoài-Minh Nguyên	17
Frédéric Hélein	13
Felix E. Browder	12
Louis Nirenberg	10
P. Mironescu	10
Jean Van Schaftingen	9
Élliott H. Lieb	7
Po‐Lam Yung	7
Itai Shafrir	7
Moshe Marcus	6
L. A. Peletier	5
Alessio Figalli	4
Andreas Seeger	4
Ron Kimmel	4
Yonathan Aflalo	4
Luigi Ambrosio	3
A. Pazy	3
Philippe Benilán	3
Yanyan Li	3
Burton H. Singer	3
Walter A. Rosenkrantz	3
Avner Friedman	3
Pierre‐Louis Lions	2
Thierry Cazenave	2
Wolfgang Arendt	2
Louis Dupaigne	2
Alfred M. Bruckstein⋆	2
Michel Chipot	2
Luc Oswald	2
Michael G. Crandall	2
Jean‐Michel Coron	2
Walter A. Strauss	2
Giandomenico Orlandi	2
Abbas Bahri	2
W. Browder	1
Stephen Wainger	1
Claude Bardos	1
Alan E. Berger	1
Nir Sochen	1
Thierry Gallouët	1
Nicola Fusco	1
Carlo Sbordone	1
Alain Haraux	1
J P Bourguignon	1
Juan Luís Vázquez	1
Alain Bensoussan	1

Lifting in Sobolev spaces

2000-12-01

Jean Bourgain , Haïm Brézis , Petru Mironescu

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Harmonic maps with defects

1986-12-01

Ha�m Brezis , Jean‐Michel Coron , Élliott H. Lieb

H1/2 maps with values into the circle: Minimal Connections, Lifting, and the Ginzburg–Landau equation

2004-06-01

Jean Bourgain , Haïm Brézis , Petru Mironescu

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Degree theory and BMO; part I: Compact manifolds without boundaries

1995-09-01

Haïm Brézis , Louis Nirenberg

Degree theory and BMO; part II: Compact manifolds with boundaries

1996-03-01

Haïm Brézis , Louis Nirenberg

Density of smooth functions between two manifolds in Sobolev spaces

1988-09-01

Fabrice Béthuel , Xiaomin Zheng

Boundary regularity and the Dirichlet problem for harmonic maps

1983-01-01

Richard Schoen , Karen Uhlenbeck

In a previous paper [10] we developed an interior regularity theory for energy minimizing harmonic maps into Riemannian manifolds. In the first two sections of this paper we prove boundary … In a previous paper [10] we developed an interior regularity theory for energy minimizing harmonic maps into Riemannian manifolds. In the first two sections of this paper we prove boundary regularity for energy minimizing maps with prescribed Dirichlet boundary condition. We show that such maps are regular in a full neighborhood of the boundary, assuming appropriate regularity on the manifolds, the boundary and the data. The reader may refer to Theorem 2.7 for a statement of the precise result. It is not surprising that the boundary regularity is actually stronger than the partial regularity we obtained for the interior. This is due to the fact that there are no nontrivial smooth harmonic maps from hemispheres S+~ which map the boundary S~~ = 9S+ to a point (I <j'<n — 2), and is analogous to the fact that in certain cases we were able to obtain complete regularity in the interior. Many authors have worked on boundary regularity for this general type of problem. We mention Hildebrandt and Widman [5] and Hamilton [4] as having obtained important results specifically for harmonic maps. Morrey had obtained the boundary regularity for domain dimension n = 2 in conjunction with his investigation of the Plateau problem in Riemannian manifolds [8]. In §3 of this paper, we observe that the direct method gives solvability of the Dirichlet problem under reasonable hypotheses on the manifolds. We give, as an application, an amusing proof of a theorem of Sacks and Uhlenbeck [9] on the existence of minimal 2-spheres representing the second homotopy group of a manifold. The same method gives smooth harmonic representations for πk(N) for a certain class of manifolds N. These are characterized by the nonexistence of lower dimensional harmonic spheres whose homogeneous extensions are minimal (see Proposition 3.4). In the last section of the paper we discuss approximation of L maps by smooth maps. We give a simple example of an L map from the three-dimensional ball to the two-sphere which is not an L limit of continuous maps. We

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Functional Analysis, Sobolev Spaces and Partial Differential Equations

2010-11-02

Haïm Brézis

Positive solutions of nonlinear elliptic equations involving critical sobolev exponents

1983-07-01

Haïm Brézis , Louis Nirenberg

Another look at Sobolev spaces

2001-01-01

Jean Bourgain , Haïm Brézis , Petru Mironescu

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A new characterization of Sobolev spaces

2006-07-03

Jean Bourgain , Hoài-Minh Nguyên

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The approximation problem for Sobolev maps between two manifolds

1991-01-01

Fabrice Béthuel

THEOREM 2. For every l~p<n,R ~ (resp.R~) is dense in WI'P(M ", Nk).We have also the following, which is the analogue of Theorem 1 bis: THEOREM 2 bis.Assume l <,p<n, … THEOREM 2. For every l~p<n,R ~ (resp.R~) is dense in WI'P(M ", Nk).We have also the following, which is the analogue of Theorem 1 bis: THEOREM 2 bis.Assume l <,p<n, and aM"~=;3.Let u be in WI'p(M n, N k) such that u restricted to OM" is in WI'v(aM ", Nk)NC ~ (resp.C=(M ", Nk)).If there is a map v in C~ ", N k) (resp.C=(M ", Nk)) such that u=v on aM", then u can be approximated in WI'p(M ", N k) by maps in R~ (resp.R~) which coincide with u on OM". --Homotopy classes in Sobolev spaces and the existence of energy minimizing maps.

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Schrödinger operators with singular potentials

1972-03-01

Tosis Kato

Some new characterizations of Sobolev spaces

2006-05-19

Hoài-Minh Nguyên

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On the structure of the Sobolev space H1/2 with values into the circle

2000-07-01

Jean Bourgain , Haïm Brézis , Petru Mironescu

On Elliptic Partial Differential Equations

2011-01-01

Louis Nirenberg

Γ-convergence and Sobolev norms

2007-11-26

Hoài-Minh Nguyên

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On functions of bounded mean oscillation

1961-08-01

Franklin John , Louis Nirenberg

Large solutions for harmonic maps in two dimensions

1983-06-01

Haïm Brézis , Jean Michel Coron

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A regularity theory for harmonic maps

1982-01-01

Richard Schoen , Karen Uhlenbeck

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How to recognize constant functions. Connections with Sobolev spaces

2002-08-31

Haïm Brézis

A criterion for a function to belong to or to is given. Various integral conditions under which a measurable function is constant are discussed. A criterion for a function to belong to or to is given. Various integral conditions under which a measurable function is constant are discussed.

A boundary value problem related to the Ginzburg-Landau model

1991-11-01

Anne Boutet de Monvel-Berthier , Vladimir Georgescu , Radu Purice

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Degree and Sobolev spaces

1999-06-01

Haïm Brézis , Yanyan Li , Petru Mironescu +1 more

Let $u$ belong (for example) to $W^{1,n+1}(S^n\times \Lambda, S^n)_{\lambda\in\Lambda}$ where $\Lambda$ is a connected open set in ${\mathbb R}^k$. For a.e. the map $x\mapsto u(x,\lambda)$ is continuous from $S^n$ into … Let $u$ belong (for example) to $W^{1,n+1}(S^n\times \Lambda, S^n)_{\lambda\in\Lambda}$ where $\Lambda$ is a connected open set in ${\mathbb R}^k$. For a.e. the map $x\mapsto u(x,\lambda)$ is continuous from $S^n$ into $S^n$ and therefore its (Brouwer) degree is well defined. We prove that this degree is independent of $\lambda$ a.e. in $\Lambda$. This result is extended to a more general setting, as well to fractional Sobolev spaces $W^{s,p}$ with $sp\geq n+1$.

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A characterization of maps in H1(B3, S2) which can be approximated by smooth maps

1990-08-01

Fabrice Béthuel

Elliptic Partial Differential Equations of Second Order

2001-01-01

David Gilbarg , Neil S. Trudinger

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Topics in Nonlinear Functional Analysis

2001-04-26

Louis Nirenberg

Topological approach: Finite dimensions Topological degree in Banach space Bifurcation theory Further topological methods Monotone operators and the min-max theorem Generalized implicit function theorems Bibliography. Topological approach: Finite dimensions Topological degree in Banach space Bifurcation theory Further topological methods Monotone operators and the min-max theorem Generalized implicit function theorems Bibliography.

Liquid Crystals and Energy Estimates for S2- Valued Maps

1987-01-01

Haïm Brézis

This report summarizes results obtained in collaboration with J.M. Coron and E. Lieb (see [3] and [4]); it answers some questions raised by J. Ericksen and D. Kinderlehrer. The original … This report summarizes results obtained in collaboration with J.M. Coron and E. Lieb (see [3] and [4]); it answers some questions raised by J. Ericksen and D. Kinderlehrer. The original motivation comes from the theory of liquid crystals (see [7], [8], [10]), and is well explained in other contributions to this volume.

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Elliptic Partial Differential Equations of Second Order

1977-01-01

David Gilbarg , Neil S. Trudinger

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Co-area, liquid crystals, and minimal surfaces

1988-01-01

F. J. Almgren , W. Browder , Élliott H. Lieb

Lifting, degree, and distributional Jacobian revisited

2004-11-02

Jean Bourgain , Haïm Brézis , Petru Mironescu

Existence and uniqueness of entropy solutions for nonlinear elliptic equations with measure data

1996-10-01

Lucio Boccardo , Thierry Gallouët , Luigi Orsina

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Relaxed Energies for Harmonic Maps

1990-01-01

Fabrice Béthuel , H. Brézis , Jean‐Michel Coron

Let % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuyQdCLaey %OGIWSaeSyhHe6aaWbaaSqabeaacaaIZaaaaaaa!3BD8 $$Omega \subset {\mathbb{R}^3}$$ be an open bounded set such that % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr … Let % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuyQdCLaey %OGIWSaeSyhHe6aaWbaaSqabeaacaaIZaaaaaaa!3BD8 $$Omega \subset {\mathbb{R}^3}$$ be an open bounded set such that % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOaIyRaeu % yQdCfaaa!38E8 $$ \partial \Omega $$ is smooth. Set % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamisamaaCa % aaleqabaGaaGymaaaakmaabmaabaGaeuyQdCLaai4oaiaadofadaah % aaWcbeqaaiaaikdaaaaakiaawIcacaGLPaaacqGH9aqpdaGadaqaai % aadwhacqGHiiIZcaWGibWaaWbaaSqabeaacaaIXaaaaOWaaeWaaeaa % cqqHPoWvcaGG7aGaeSyhHe6aaWbaaSqabeaacaaIZaaaaaGccaGLOa % GaayzkaaGaai4oaiaacYhacaWG1bWaaeWaaeaacaWG4baacaGLOaGa % ayzkaaGaaiiFaiabg2da9iaaigdaieGacaWFGaGaa8xyaiaa-5caca % WFLbGaa8NlaaGaay5Eaiaaw2haaaaa!56D4 $$ {H^1}\left( {\Omega ;{S^2}} \right) = \left\{ {u \in {H^1}\left( {\Omega ;{\mathbb{R}^3}} \right);|u\left( x \right)| = 1 a.e.} \right\} $$ and % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamisamaaDa % aaleaacqaHgpGAaeaacaaIXaaaaOWaaeWaaeaacqqHPoWvcaGG7aGa % am4uamaaCaaaleqabaGaaGOmaaaaaOGaayjkaiaawMcaaiabg2da9m % aacmaabaGaamyDaiabgIGiolaadIeadaahaaWcbeqaaiaaigdaaaGc % daqadaqaaiabfM6axjaacUdacaWGtbWaaWbaaSqabeaacaaIYaaaaa % GccaGLOaGaayzkaaGaai4oaiaadwhacqGH9aqpcqaHgpGAieGacaWF % GaGaa83Baiaa-5gacaWFGaGaeyOaIyRaeuyQdCfacaGL7bGaayzFaa % Gaaiilaaaa!5772 $$ H_\varphi ^1\left( {\Omega ;{S^2}} \right) = \left\{ {u \in {H^1}\left( {\Omega ;{S^2}} \right);u = \varphi on \partial \Omega } \right\}, $$ where % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOXdOMaai % OoaiabgkGi2kabfM6axjabgkziUkaadofadaahaaWcbeqaaiaaikda % aaaaaa!3F11 $$ \varphi :\partial \Omega \to {S^2} $$ is a given boundary data.

Gagliardo-Nirenberg, composition and products in fractional Sobolev spaces

2001-12-01

Haïm Brézis , Petru Mironescu

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Semi-linear second-order elliptic equations in L1

1973-10-01

Haïm Brézis , Walter A. Strauss

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Γ-convergence, Sobolev norms, and BV functions

2011-04-01

Hoài-Minh Nguyên

We prove that the family of functionals (Iδ) defined by Iδ(g)=∫∫RN×RN|g(x)-g(y)|>δδp|x-y|N+pdxdy, ∀g∈Lp(RN), for p≥1 and δ>0, Γ-converges in Lp(RN), as δ goes to zero, when p≥1. Hereafter | | denotes … We prove that the family of functionals (Iδ) defined by Iδ(g)=∫∫RN×RN|g(x)-g(y)|>δδp|x-y|N+pdxdy, ∀g∈Lp(RN), for p≥1 and δ>0, Γ-converges in Lp(RN), as δ goes to zero, when p≥1. Hereafter | | denotes the Euclidean norm of RN. We also introduce a characterization for bounded variation (BV) functions which has some advantages in comparison with the classic one based on the notion of essential variation on almost every line.

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Some remarks on the density of regular mappings in Sobolev classes of SM. Values functions

1988-01-01

Miguel Escobedo

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Some inequalities related to Sobolev norms

2010-10-12

Hoài-Minh Nguyên

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Complex Ginzburg-Landau equations in high dimensions and codimension two area minimizing currents

1999-09-30

Fanghua Lin , Tristan Rivière

Abstract. There is an obvious topological obstruction for a finite energy unimodular harmonic extension of a S^1 -valued function defined on the boundary of a bounded regular domain of R^n … Abstract. There is an obvious topological obstruction for a finite energy unimodular harmonic extension of a S^1 -valued function defined on the boundary of a bounded regular domain of R^n . When such extensions do not exist, we use the Ginzburg–Landau relaxation procedure. We prove that, up to a subsequence, a sequence of Ginzburg–Landau minimizers, as the coupling parameter tends to infinity, converges to a unimodular harmonic map away from a codimension-2 minimal current minimizing the area within the homology class induced from the S^1 -valued boundary data. The union of this harmonic map and the minimal current is the natural generalization of the harmonic extension.

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Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities

2002-01-01

Élliott H. Lieb

Best constant in Sobolev inequality

1976-12-01

Giorgio Talenti

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Multiple Integrals in the Calculus of Variations

1966-01-01

Charles B. Morrey

Measure Theory and Fine Properties of Functions

2018-04-27

Lawrence C. Evans , Ronald F. Garzepy

This book provides a detailed examination of the central assertions of measure theory in n-dimensional Euclidean space and emphasizes the roles of Hausdorff measure and the capacity in characterizing the … This book provides a detailed examination of the central assertions of measure theory in n-dimensional Euclidean space and emphasizes the roles of Hausdorff measure and the capacity in characterizing the fine properties of sets and functions. Topics covered include a quick review of abstract measure theory, theorems and differentiation in Mn, lower Hausdorff measures, area and coarea formulas for Lipschitz mappings and related change-of-variable formulas, and Sobolev functions and functions of bounded variation. The text provides complete proofs of many key results omitted from other books, including Besicovitch's Covering Theorem, Rademacher's Theorem (on the differentiability a.e. of Lipschitz functions), the Area and Coarea Formulas, the precise structure of Sobolev and BV functions, the precise structure of sets of finite perimeter, and Alexandro's Theorem (on the twice differentiability a.e. of convex functions).Topics are carefully selected and the proofs succinct, but complete, which makes this book ideal reading for applied mathematicians and graduate students in applied mathematics.

Topology and Sobolev Spaces

2001-07-01

Haïm Brézis , Yanyan Li

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Geometric Measure Theory

1988-01-01

Herbert Fédérer

Operateurs Maximaux Monotones - Et Semi-Groupes De Contractions Dans Les Espaces De Hilbert

1973-01-01

Haïm Brézis

BMO‐Type Norms Related to the Perimeter of Sets

2015-10-05

Luigi Ambrosio , Jean Bourgain , Haïm Brézis +1 more

Abstract In this paper we consider an isotropic variant of the BMO‐type norm recently introduced (Bourgain, Brezis, and Mironescu, 2015). We prove that, when considering characteristic functions of sets, this … Abstract In this paper we consider an isotropic variant of the BMO‐type norm recently introduced (Bourgain, Brezis, and Mironescu, 2015). We prove that, when considering characteristic functions of sets, this norm is related to the perimeter. A byproduct of our analysis is a new characterization of the perimeter of sets in terms of this norm, independent of the theory of distributions.© 2016 Wiley Periodicals, Inc.

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Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations

1996-12-31

Thomas Runst , Winfried Sickel

Further characterizations of Sobolev spaces

2008-03-31

Hoài-Minh Nguyên

Let (F n ) n∈N be a sequence of non-decreasing functions from [0, +∞) into [0, +∞).Under some suitable hypotheses onwhere K N,p is a positive constant depending only on … Let (F n ) n∈N be a sequence of non-decreasing functions from [0, +∞) into [0, +∞).Under some suitable hypotheses onwhere K N,p is a positive constant depending only on N and p.This extends some results in

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Équations et inéquations non linéaires dans les espaces vectoriels en dualité

1968-01-01

Haïm Brézis

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Singularities of positive supersolutions in elliptic PDEs

2004-11-01

Louis Dupaigne , Augusto C. Ponce

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