Motivated by a recent study of Bessel operators in connection with a refinement of Hardy's inequality involving $ 1/\sin^2(x) $ on the finite interval $ (0, \pi) $, we now …
Motivated by a recent study of Bessel operators in connection with a refinement of Hardy's inequality involving $ 1/\sin^2(x) $ on the finite interval $ (0, \pi) $, we now take a closer look at the underlying Bessel-type operators with more general inverse square singularities at the interval endpoints. More precisely, we consider quadratic forms and operator realizations in $ L^2((a, b); dx) $ associated with differential expressions of the form \begin{document}$ \omega_{s_a} = - \frac{d^2}{dx^2} + \frac{s_a^2 - (1/4)}{(x-a)^2}, \quad s_a \in {\mathbb{R}}, \; x \in (a, b), $\end{document} and$ \begin{align*} \tau_{s_a, s_b} = - \frac{d^2}{dx^2} + \frac{s_a^2 - (1/4)}{(x-a)^2} + \frac{s_b^2 - (1/4)}{(x-b)^2} + q(x), \quad x \in (a, b), & \\ s_a, s_b \in [0, \infty), \; q \in L^{\infty}((a, b); dx), \; q \text{ real-valued a.e. on } (a, b) , & \end{align*} $where $ (a, b) \subset {\mathbb{R}} $ is a bounded interval.As an explicit illustration we describe the Krein–von Neumann extension of the minimal operator corresponding to $ \omega_{s_a} $ and $ \tau_{s_a, s_b} $.
In this paper we prove that the level sets of the first non-constant eigenfunction of the Neumann Laplacian on a convex planar domain have only finitely many connected components. This …
In this paper we prove that the level sets of the first non-constant eigenfunction of the Neumann Laplacian on a convex planar domain have only finitely many connected components. This problem is motivated, in part, by the hot spots conjecture of J. Rauch.
Abstract The principal purpose of this note is to prove a logarithmic refinement of the power weighted Hardy–Rellich inequality on n -dimensional balls, valid for the largest variety of underlying …
Abstract The principal purpose of this note is to prove a logarithmic refinement of the power weighted Hardy–Rellich inequality on n -dimensional balls, valid for the largest variety of underlying parameters and for all dimensions $$n \in {\mathbb {N}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>N</mml:mi> </mml:mrow> </mml:math> , $$n\ge 2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> .
The principal aim of this paper is to prove the inequality \begin{align*} \int_0^\pi dx \, |f'(x)|^2 \geq \dfrac{1}{4}\int_0^\pi dx \, \dfrac{|f(x)|^2}{\sin^2 (x)}+\dfrac{1}{4}\int_0^\pi dx \, |f(x)|^2,\quad f\in H_0^1 ((0,\pi)), \end{align*} where …
The principal aim of this paper is to prove the inequality \begin{align*} \int_0^\pi dx \, |f'(x)|^2 \geq \dfrac{1}{4}\int_0^\pi dx \, \dfrac{|f(x)|^2}{\sin^2 (x)}+\dfrac{1}{4}\int_0^\pi dx \, |f(x)|^2,\quad f\in H_0^1 ((0,\pi)), \end{align*} where both constants $1/4$ appearing in the above inequality are optimal. In addition, this inequality is strict in the sense that equality holds if and only if $f \equiv 0$. This inequality is derived with the help of the exactly solvable, strongly singular, Dirichlet-type Schrodinger operator associated with the differential expression \begin{align*} \tau_s=-\dfrac{d^2}{dx^2}+\dfrac{s^2-(1/4)}{\sin^2 (x)}, \quad s \in [0,\infty), \; x \in (0,\pi). \end{align*}
The new inequality represents a refinement of Hardy's classical inequality \begin{align*} \int_0^\pi dx \, |f'(x)|^2 \geq \dfrac{1}{4}\int_0^\pi dx \, \dfrac{|f(x)|^2}{x^2}, \quad f\in H_0^1 ((0,\pi)), \end{align*} it also improves upon one of its well-known extensions in the form \begin{align*} \int_0^\pi dx \, |f'(x)|^2 \geq \dfrac{1}{4}\int_0^\pi dx \, \dfrac{|f(x)|^2}{d_{(0,\pi)}(x)^2}, \quad f\in H_0^1 ((0,\pi)), \end{align*} where $d_{(0,\pi)}(x)$ represents the distance from $x \in (0,\pi)$ to the boundary $\{0,\pi\}$ of $(0,\pi)$.
In addition, we hint at the possibility to extend this inequality to more general situations where the differential expression is of the form $\tau = -(d^2/dx^2) + q(x)$, $x \in (a,b) \subset \mathbb{R}$, where $q(\, \cdot \,)$ behaves like $C_c (x-c)^{-2}$ near $x=c$, $c \in \{a,b\}$, with $C_c\geq - 1/4$.
We revisit an upper heat kernel bound for second order uniformly elliptic operators H defined on bounded regions Ω in R N .This bound is of the typewhere E 1 …
We revisit an upper heat kernel bound for second order uniformly elliptic operators H defined on bounded regions Ω in R N .This bound is of the typewhere E 1 and φ 1 are, respectively, the ground state eigenvalue and the normalized ground state eigenfunction of H , Λ is the upper ellipticity constant of H , a > 0 is a constant related to a lower bound of φ 1 near the boundary ∂ Ω , and c 1 > 0 is a constant which depends on Ω , E 1 , the ellipticity constants of H , and a lower bound of φ 1 near ∂ Ω .In particular, this bound provides a corrected version of a bound originally studied in [2] for large time t > 0 .
We revisit weighted Hardy-type inequalities employing an elementary ad hoc approach that yields explicit constants. We also discuss the infinite sequence of power weighted Birman-Hardy-Rellich-type inequalities and derive an operator-valued …
We revisit weighted Hardy-type inequalities employing an elementary ad hoc approach that yields explicit constants. We also discuss the infinite sequence of power weighted Birman-Hardy-Rellich-type inequalities and derive an operator-valued version thereof.
The principal aim of this paper is to establish the optimality (i.e., sharpness) of the constants A(m, α) and B(m, α), m â â, α â â, of the form
The principal aim of this paper is to establish the optimality (i.e., sharpness) of the constants A(m, α) and B(m, α), m â â, α â â, of the form
We revisit weighted Hardy-type inequalities employing an elementary ad hoc approach that yields explicit constants.We also discuss the infinite sequence of power weighted Birman-Hardy-Rellich-type inequalities and derive an operator-valued version …
We revisit weighted Hardy-type inequalities employing an elementary ad hoc approach that yields explicit constants.We also discuss the infinite sequence of power weighted Birman-Hardy-Rellich-type inequalities and derive an operator-valued version thereof.
Motivated by a recent study of Bessel operators in connection with a refinement of Hardy's inequality involving $1/\sin^2(x)$ on the finite interval $(0,\pi)$, we now take a closer look at …
Motivated by a recent study of Bessel operators in connection with a refinement of Hardy's inequality involving $1/\sin^2(x)$ on the finite interval $(0,\pi)$, we now take a closer look at the underlying Bessel-type operators with more general inverse square singularities at the interval endpoints. More precisely, we consider quadratic forms and operator realizations in $L^2((a,b); dx)$ associated with differential expressions of the form \[ \omega_{s_a} = - \frac{d^2}{dx^2} + \frac{s_a^2 - (1/4)}{(x-a)^2}, \quad s_a \in \mathbb{R}, \; x \in (a,b), \] and \begin{align*} \tau_{s_a,s_b} = - \frac{d^2}{dx^2} + \frac{s_a^2 - (1/4)}{(x-a)^2} + \frac{s_b^2 - (1/4)}{(x-b)^2} + q(x), \quad x \in (a,b),& \\ s_a, s_b \in [0,\infty), \; q \in L^{\infty}((a,b); dx), \; q \text{ real-valued~a.e.~on $(a,b)$,}& \end{align*} where $(a,b) \subset \mathbb{R}$ is a bounded interval. As an explicit illustration we describe the Krein-von Neumann extension of the minimal operator corresponding $\omega_{s_a}$ and $\tau_{s_a,s_b}$.
The principal aim of this paper is to employ Bessel-type operators in proving the inequality \begin{align*} \int_0^\pi dx \, |f'(x)|^2 \geq \dfrac{1}{4}\int_0^\pi dx \, \dfrac{|f(x)|^2}{\sin^2 (x)}+\dfrac{1}{4}\int_0^\pi dx \, |f(x)|^2,\quad f\in …
The principal aim of this paper is to employ Bessel-type operators in proving the inequality \begin{align*} \int_0^\pi dx \, |f'(x)|^2 \geq \dfrac{1}{4}\int_0^\pi dx \, \dfrac{|f(x)|^2}{\sin^2 (x)}+\dfrac{1}{4}\int_0^\pi dx \, |f(x)|^2,\quad f\in H_0^1 ((0,\pi)), \end{align*} where both constants $1/4$ appearing in the above inequality are optimal. In addition, this inequality is strict in the sense that equality holds if and only if $f \equiv 0$. This inequality is derived with the help of the exactly solvable, strongly singular, Dirichlet-type Schr\"{o}dinger operator associated with the differential expression \begin{align*} \tau_s=-\dfrac{d^2}{dx^2}+\dfrac{s^2-(1/4)}{\sin^2 (x)}, \quad s \in [0,\infty), \; x \in (0,\pi). \end{align*} The new inequality represents a refinement of Hardy's classical inequality \begin{align*} \int_0^\pi dx \, |f'(x)|^2 \geq \dfrac{1}{4}\int_0^\pi dx \, \dfrac{|f(x)|^2}{x^2}, \quad f\in H_0^1 ((0,\pi)), \end{align*} it also improves upon one of its well-known extensions in the form \begin{align*} \int_0^\pi dx \, |f'(x)|^2 \geq \dfrac{1}{4}\int_0^\pi dx \, \dfrac{|f(x)|^2}{d_{(0,\pi)}(x)^2}, \quad f\in H_0^1 ((0,\pi)), \end{align*} where $d_{(0,\pi)}(x)$ represents the distance from $x \in (0,\pi)$ to the boundary $\{0,\pi\}$ of $(0,\pi)$.
The principal aim of this paper is to extend Birman's sequence of integral inequalities originally obtained in 1961, and containing Hardy's and Rellich's inequality as special cases, to a sequence …
The principal aim of this paper is to extend Birman's sequence of integral inequalities originally obtained in 1961, and containing Hardy's and Rellich's inequality as special cases, to a sequence of inequalities that incorporates power weights on either side and logarithmic refinements on the right-hand side of the inequality as well. Our new technique of proof for this sequence of inequalities relies on a combination of transforms originally due to Hartman and M\"uller-Pfeiffer. The results obtained considerably improve on prior results in the literature.
We revisit weighted Hardy-type inequalities employing an elementary ad hoc approach that yields explicit constants. We also discuss the infinite sequence of power weighted Birman-Hardy-Rellich-type inequalities and derive an operator-valued …
We revisit weighted Hardy-type inequalities employing an elementary ad hoc approach that yields explicit constants. We also discuss the infinite sequence of power weighted Birman-Hardy-Rellich-type inequalities and derive an operator-valued version thereof.
The principal purpose of this note is to prove a logarithmic refinement of the power weighted Hardy--Rellich inequality on $n$-dimensional balls, valid for the largest variety of underlying parameters and …
The principal purpose of this note is to prove a logarithmic refinement of the power weighted Hardy--Rellich inequality on $n$-dimensional balls, valid for the largest variety of underlying parameters and for all dimensions $n \in \mathbb{N}$, $n\geq 2$.
We revisit and extend a variety of inequalities related to power weighted Rellich and Hardy--Rellich inequalities, including an inequality due to Schmincke.
We revisit and extend a variety of inequalities related to power weighted Rellich and Hardy--Rellich inequalities, including an inequality due to Schmincke.
We derive an optimal power-weighted Hardy-type inequality in integral form on finite intervals and subsequently prove the analogous inequality in differential form. We note that the optimal constant of the …
We derive an optimal power-weighted Hardy-type inequality in integral form on finite intervals and subsequently prove the analogous inequality in differential form. We note that the optimal constant of the latter inequality differs from the former. Moreover, by iterating these inequalities we derive the sequence of power-weighted Birman-Hardy-Rellich-type inequalities in integral form on finite intervals and then also prove the analogous sequence of inequalities in differential form. We use the one-dimensional Hardy-type result in differential form to derive an optimal multi-dimensional version of the power-weighted Hardy inequality in differential form on annuli (i.e., spherical shell domains), and once more employ an iteration procedure to derive the Birman-Hardy-Rellich-type sequence of power-weighted higher-order Hardy-type inequalities for annuli. In the limit as the annulus approaches $\mathbb{R}^n$, we recover well-known prior results on Rellich-type inequalities on $\mathbb{R}^n$.
Abstract We derive an optimal power‐weighted Hardy‐type inequality in integral form on finite intervals and subsequently prove the analogous inequality in differential form. We note that the optimal constant of …
Abstract We derive an optimal power‐weighted Hardy‐type inequality in integral form on finite intervals and subsequently prove the analogous inequality in differential form. We note that the optimal constant of the latter inequality differs from the former. Moreover, by iterating these inequalities we derive the sequence of power‐weighted Birman–Hardy–Rellich‐type inequalities in integral form on finite intervals and then also prove the analogous sequence of inequalities in differential form. We use the one‐dimensional Hardy‐type result in differential form to derive an optimal multi‐dimensional version of the power‐weighted Hardy inequality in differential form on annuli (i.e., spherical shell domains), and once more employ an iteration procedure to derive the Birman–Hardy–Rellich‐type sequence of power‐weighted higher order Hardy‐type inequalities for annuli. In the limit as the annulus approaches \{0}, we recover well‐known prior results on Rellich‐type inequalities on \{0}.
Abstract We derive an optimal power‐weighted Hardy‐type inequality in integral form on finite intervals and subsequently prove the analogous inequality in differential form. We note that the optimal constant of …
Abstract We derive an optimal power‐weighted Hardy‐type inequality in integral form on finite intervals and subsequently prove the analogous inequality in differential form. We note that the optimal constant of the latter inequality differs from the former. Moreover, by iterating these inequalities we derive the sequence of power‐weighted Birman–Hardy–Rellich‐type inequalities in integral form on finite intervals and then also prove the analogous sequence of inequalities in differential form. We use the one‐dimensional Hardy‐type result in differential form to derive an optimal multi‐dimensional version of the power‐weighted Hardy inequality in differential form on annuli (i.e., spherical shell domains), and once more employ an iteration procedure to derive the Birman–Hardy–Rellich‐type sequence of power‐weighted higher order Hardy‐type inequalities for annuli. In the limit as the annulus approaches \{0}, we recover well‐known prior results on Rellich‐type inequalities on \{0}.
Abstract The principal purpose of this note is to prove a logarithmic refinement of the power weighted Hardy–Rellich inequality on n -dimensional balls, valid for the largest variety of underlying …
Abstract The principal purpose of this note is to prove a logarithmic refinement of the power weighted Hardy–Rellich inequality on n -dimensional balls, valid for the largest variety of underlying parameters and for all dimensions $$n \in {\mathbb {N}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>N</mml:mi> </mml:mrow> </mml:math> , $$n\ge 2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> .
We derive an optimal power-weighted Hardy-type inequality in integral form on finite intervals and subsequently prove the analogous inequality in differential form. We note that the optimal constant of the …
We derive an optimal power-weighted Hardy-type inequality in integral form on finite intervals and subsequently prove the analogous inequality in differential form. We note that the optimal constant of the latter inequality differs from the former. Moreover, by iterating these inequalities we derive the sequence of power-weighted Birman-Hardy-Rellich-type inequalities in integral form on finite intervals and then also prove the analogous sequence of inequalities in differential form. We use the one-dimensional Hardy-type result in differential form to derive an optimal multi-dimensional version of the power-weighted Hardy inequality in differential form on annuli (i.e., spherical shell domains), and once more employ an iteration procedure to derive the Birman-Hardy-Rellich-type sequence of power-weighted higher-order Hardy-type inequalities for annuli. In the limit as the annulus approaches $\mathbb{R}^n$, we recover well-known prior results on Rellich-type inequalities on $\mathbb{R}^n$.
We revisit and extend a variety of inequalities related to power weighted Rellich and Hardy--Rellich inequalities, including an inequality due to Schmincke.
We revisit and extend a variety of inequalities related to power weighted Rellich and Hardy--Rellich inequalities, including an inequality due to Schmincke.
The principal purpose of this note is to prove a logarithmic refinement of the power weighted Hardy--Rellich inequality on $n$-dimensional balls, valid for the largest variety of underlying parameters and …
The principal purpose of this note is to prove a logarithmic refinement of the power weighted Hardy--Rellich inequality on $n$-dimensional balls, valid for the largest variety of underlying parameters and for all dimensions $n \in \mathbb{N}$, $n\geq 2$.
Motivated by a recent study of Bessel operators in connection with a refinement of Hardy's inequality involving $ 1/\sin^2(x) $ on the finite interval $ (0, \pi) $, we now …
Motivated by a recent study of Bessel operators in connection with a refinement of Hardy's inequality involving $ 1/\sin^2(x) $ on the finite interval $ (0, \pi) $, we now take a closer look at the underlying Bessel-type operators with more general inverse square singularities at the interval endpoints. More precisely, we consider quadratic forms and operator realizations in $ L^2((a, b); dx) $ associated with differential expressions of the form \begin{document}$ \omega_{s_a} = - \frac{d^2}{dx^2} + \frac{s_a^2 - (1/4)}{(x-a)^2}, \quad s_a \in {\mathbb{R}}, \; x \in (a, b), $\end{document} and$ \begin{align*} \tau_{s_a, s_b} = - \frac{d^2}{dx^2} + \frac{s_a^2 - (1/4)}{(x-a)^2} + \frac{s_b^2 - (1/4)}{(x-b)^2} + q(x), \quad x \in (a, b), & \\ s_a, s_b \in [0, \infty), \; q \in L^{\infty}((a, b); dx), \; q \text{ real-valued a.e. on } (a, b) , & \end{align*} $where $ (a, b) \subset {\mathbb{R}} $ is a bounded interval.As an explicit illustration we describe the Krein–von Neumann extension of the minimal operator corresponding to $ \omega_{s_a} $ and $ \tau_{s_a, s_b} $.
The principal aim of this paper is to establish the optimality (i.e., sharpness) of the constants A(m, α) and B(m, α), m â â, α â â, of the form
The principal aim of this paper is to establish the optimality (i.e., sharpness) of the constants A(m, α) and B(m, α), m â â, α â â, of the form
The principal aim of this paper is to prove the inequality \begin{align*} \int_0^\pi dx \, |f'(x)|^2 \geq \dfrac{1}{4}\int_0^\pi dx \, \dfrac{|f(x)|^2}{\sin^2 (x)}+\dfrac{1}{4}\int_0^\pi dx \, |f(x)|^2,\quad f\in H_0^1 ((0,\pi)), \end{align*} where …
The principal aim of this paper is to prove the inequality \begin{align*} \int_0^\pi dx \, |f'(x)|^2 \geq \dfrac{1}{4}\int_0^\pi dx \, \dfrac{|f(x)|^2}{\sin^2 (x)}+\dfrac{1}{4}\int_0^\pi dx \, |f(x)|^2,\quad f\in H_0^1 ((0,\pi)), \end{align*} where both constants $1/4$ appearing in the above inequality are optimal. In addition, this inequality is strict in the sense that equality holds if and only if $f \equiv 0$. This inequality is derived with the help of the exactly solvable, strongly singular, Dirichlet-type Schrodinger operator associated with the differential expression \begin{align*} \tau_s=-\dfrac{d^2}{dx^2}+\dfrac{s^2-(1/4)}{\sin^2 (x)}, \quad s \in [0,\infty), \; x \in (0,\pi). \end{align*}
The new inequality represents a refinement of Hardy's classical inequality \begin{align*} \int_0^\pi dx \, |f'(x)|^2 \geq \dfrac{1}{4}\int_0^\pi dx \, \dfrac{|f(x)|^2}{x^2}, \quad f\in H_0^1 ((0,\pi)), \end{align*} it also improves upon one of its well-known extensions in the form \begin{align*} \int_0^\pi dx \, |f'(x)|^2 \geq \dfrac{1}{4}\int_0^\pi dx \, \dfrac{|f(x)|^2}{d_{(0,\pi)}(x)^2}, \quad f\in H_0^1 ((0,\pi)), \end{align*} where $d_{(0,\pi)}(x)$ represents the distance from $x \in (0,\pi)$ to the boundary $\{0,\pi\}$ of $(0,\pi)$.
In addition, we hint at the possibility to extend this inequality to more general situations where the differential expression is of the form $\tau = -(d^2/dx^2) + q(x)$, $x \in (a,b) \subset \mathbb{R}$, where $q(\, \cdot \,)$ behaves like $C_c (x-c)^{-2}$ near $x=c$, $c \in \{a,b\}$, with $C_c\geq - 1/4$.
Motivated by a recent study of Bessel operators in connection with a refinement of Hardy's inequality involving $1/\sin^2(x)$ on the finite interval $(0,\pi)$, we now take a closer look at …
Motivated by a recent study of Bessel operators in connection with a refinement of Hardy's inequality involving $1/\sin^2(x)$ on the finite interval $(0,\pi)$, we now take a closer look at the underlying Bessel-type operators with more general inverse square singularities at the interval endpoints. More precisely, we consider quadratic forms and operator realizations in $L^2((a,b); dx)$ associated with differential expressions of the form \[ \omega_{s_a} = - \frac{d^2}{dx^2} + \frac{s_a^2 - (1/4)}{(x-a)^2}, \quad s_a \in \mathbb{R}, \; x \in (a,b), \] and \begin{align*} \tau_{s_a,s_b} = - \frac{d^2}{dx^2} + \frac{s_a^2 - (1/4)}{(x-a)^2} + \frac{s_b^2 - (1/4)}{(x-b)^2} + q(x), \quad x \in (a,b),& \\ s_a, s_b \in [0,\infty), \; q \in L^{\infty}((a,b); dx), \; q \text{ real-valued~a.e.~on $(a,b)$,}& \end{align*} where $(a,b) \subset \mathbb{R}$ is a bounded interval. As an explicit illustration we describe the Krein-von Neumann extension of the minimal operator corresponding $\omega_{s_a}$ and $\tau_{s_a,s_b}$.
The principal aim of this paper is to employ Bessel-type operators in proving the inequality \begin{align*} \int_0^\pi dx \, |f'(x)|^2 \geq \dfrac{1}{4}\int_0^\pi dx \, \dfrac{|f(x)|^2}{\sin^2 (x)}+\dfrac{1}{4}\int_0^\pi dx \, |f(x)|^2,\quad f\in …
The principal aim of this paper is to employ Bessel-type operators in proving the inequality \begin{align*} \int_0^\pi dx \, |f'(x)|^2 \geq \dfrac{1}{4}\int_0^\pi dx \, \dfrac{|f(x)|^2}{\sin^2 (x)}+\dfrac{1}{4}\int_0^\pi dx \, |f(x)|^2,\quad f\in H_0^1 ((0,\pi)), \end{align*} where both constants $1/4$ appearing in the above inequality are optimal. In addition, this inequality is strict in the sense that equality holds if and only if $f \equiv 0$. This inequality is derived with the help of the exactly solvable, strongly singular, Dirichlet-type Schr\"{o}dinger operator associated with the differential expression \begin{align*} \tau_s=-\dfrac{d^2}{dx^2}+\dfrac{s^2-(1/4)}{\sin^2 (x)}, \quad s \in [0,\infty), \; x \in (0,\pi). \end{align*} The new inequality represents a refinement of Hardy's classical inequality \begin{align*} \int_0^\pi dx \, |f'(x)|^2 \geq \dfrac{1}{4}\int_0^\pi dx \, \dfrac{|f(x)|^2}{x^2}, \quad f\in H_0^1 ((0,\pi)), \end{align*} it also improves upon one of its well-known extensions in the form \begin{align*} \int_0^\pi dx \, |f'(x)|^2 \geq \dfrac{1}{4}\int_0^\pi dx \, \dfrac{|f(x)|^2}{d_{(0,\pi)}(x)^2}, \quad f\in H_0^1 ((0,\pi)), \end{align*} where $d_{(0,\pi)}(x)$ represents the distance from $x \in (0,\pi)$ to the boundary $\{0,\pi\}$ of $(0,\pi)$.
We revisit weighted Hardy-type inequalities employing an elementary ad hoc approach that yields explicit constants.We also discuss the infinite sequence of power weighted Birman-Hardy-Rellich-type inequalities and derive an operator-valued version …
We revisit weighted Hardy-type inequalities employing an elementary ad hoc approach that yields explicit constants.We also discuss the infinite sequence of power weighted Birman-Hardy-Rellich-type inequalities and derive an operator-valued version thereof.
The principal aim of this paper is to extend Birman's sequence of integral inequalities originally obtained in 1961, and containing Hardy's and Rellich's inequality as special cases, to a sequence …
The principal aim of this paper is to extend Birman's sequence of integral inequalities originally obtained in 1961, and containing Hardy's and Rellich's inequality as special cases, to a sequence of inequalities that incorporates power weights on either side and logarithmic refinements on the right-hand side of the inequality as well. Our new technique of proof for this sequence of inequalities relies on a combination of transforms originally due to Hartman and M\"uller-Pfeiffer. The results obtained considerably improve on prior results in the literature.
We revisit weighted Hardy-type inequalities employing an elementary ad hoc approach that yields explicit constants. We also discuss the infinite sequence of power weighted Birman-Hardy-Rellich-type inequalities and derive an operator-valued …
We revisit weighted Hardy-type inequalities employing an elementary ad hoc approach that yields explicit constants. We also discuss the infinite sequence of power weighted Birman-Hardy-Rellich-type inequalities and derive an operator-valued version thereof.
We revisit weighted Hardy-type inequalities employing an elementary ad hoc approach that yields explicit constants. We also discuss the infinite sequence of power weighted Birman-Hardy-Rellich-type inequalities and derive an operator-valued …
We revisit weighted Hardy-type inequalities employing an elementary ad hoc approach that yields explicit constants. We also discuss the infinite sequence of power weighted Birman-Hardy-Rellich-type inequalities and derive an operator-valued version thereof.
We revisit an upper heat kernel bound for second order uniformly elliptic operators H defined on bounded regions Ω in R N .This bound is of the typewhere E 1 …
We revisit an upper heat kernel bound for second order uniformly elliptic operators H defined on bounded regions Ω in R N .This bound is of the typewhere E 1 and φ 1 are, respectively, the ground state eigenvalue and the normalized ground state eigenfunction of H , Λ is the upper ellipticity constant of H , a > 0 is a constant related to a lower bound of φ 1 near the boundary ∂ Ω , and c 1 > 0 is a constant which depends on Ω , E 1 , the ellipticity constants of H , and a lower bound of φ 1 near ∂ Ω .In particular, this bound provides a corrected version of a bound originally studied in [2] for large time t > 0 .
In this paper we prove that the level sets of the first non-constant eigenfunction of the Neumann Laplacian on a convex planar domain have only finitely many connected components. This …
In this paper we prove that the level sets of the first non-constant eigenfunction of the Neumann Laplacian on a convex planar domain have only finitely many connected components. This problem is motivated, in part, by the hot spots conjecture of J. Rauch.
Part 1 The one-dimensional Hardy inequality on (a,b): historical remarks various methods of deriving it necessary and sufficient conditions for various classes of smooth functions u (u(a) = 0, or …
Part 1 The one-dimensional Hardy inequality on (a,b): historical remarks various methods of deriving it necessary and sufficient conditions for various classes of smooth functions u (u(a) = 0, or u(b) = 0, or u(a) = u(b) = 0). Part 2 The n-dimensional Hardy inequality: the approach via the one-dimensional case (special coordinates, special weights) the approach via differential equations (general weights) the approach via weighted norm inequalities (weights from the Muckenhoupt class AP) general necessary and sufficient conditions (the Maz'ja approach via capacities). Part 3 Weighted norm inequalities (a short survey of some special results). Part 4 Imbedding theorems for weighted Sobolev spaces: the Hardy inequality as an imbedding of a weighted Sobolev space into a weighted Lebesgue space the one-dimensional case the n-dimensional case special weights.
Abstract This book gives an account of those parts of the analysis of closed linear operators acting in Banach or Hilbert spaces that are relevant to spectral problems involving differential …
Abstract This book gives an account of those parts of the analysis of closed linear operators acting in Banach or Hilbert spaces that are relevant to spectral problems involving differential operators, and makes applications to such questions. After the exposition of the abstract theory in the first four chapters, Sobolev spaces are introduced and their main properties established. The remaining seven chapters are largely concerned with second-order elliptic differential operators and related boundary-value problems. Particular attention is paid to the spectrum of the Schrödinger operator. Its original form contains material of lasting importance that is relatively unaffected by advances in the theory since 1987, when the book was first published. The present edition differs from the old by virtue of the correction of minor errors and improvements of various proofs. In addition, it contains Notes at the ends of most chapters, intended to give the reader some idea of recent developments together with additional references that enable more detailed accounts to be accessed.
The two related one space dimensional singular linear parabolic equations (1), (2) studied by H. Brezis et al. [Comm. Pure Appl. Math. 24 (1971), pp. 395â416] have different scaling properties. …
The two related one space dimensional singular linear parabolic equations (1), (2) studied by H. Brezis et al. [Comm. Pure Appl. Math. 24 (1971), pp. 395â416] have different scaling properties. These scaling properties lead to new variants of the Hardy and Caffarelli-Kohn-Nirenberg inequalities. These results are proved, and they imply some non-wellposedness results when the constant in the singular potential term is large enough.
The sharp constants in Hardy type inequalities are known only in a few cases.In this paper we discuss some situations when such sharp constants are known, but also some new …
The sharp constants in Hardy type inequalities are known only in a few cases.In this paper we discuss some situations when such sharp constants are known, but also some new sharp constants are derived both in one-dimensional and multi-dimensional cases.
Hardy type inequalities: Bessel pairs and Sturm's oscillation theory The classical Hardy inequality and its improvements Improved Hardy inequality with boundary singularity Weighted Hardy inequalities The Hardy inequality and second …
Hardy type inequalities: Bessel pairs and Sturm's oscillation theory The classical Hardy inequality and its improvements Improved Hardy inequality with boundary singularity Weighted Hardy inequalities The Hardy inequality and second order nonlinear eigenvalue problems Hardy-Rellich type inequalities: Improved Hardy-Rellich inequalities on $H^2_0(\Omega)$ Weighted Hardy-Rellich inequalities on $H^2(\Omega)\cap H^1_0(\Omega)$ Critical dimensions for $4^{\textrm{th}}$ order nonlinear eigenvalue problems Hardy inequalities for general elliptic operators: General Hardy inequalities Improved Hardy inequalities for general elliptic operators Regularity and stability of solutions in non-self-adjoint problems Mass transport and optimal geometric inequalities: A general comparison principle for interacting gases Optimal Euclidean Sobolev inequalities Geometric inequalities Hardy-Rellich-Sobolev inequalities: The Hardy-Sobolev inequalities Domain curvature and best constants in the Hardy-Sobolev inequalities Aubin-Moser-Onofri inequalities: Log-Sobolev inequalities on the real line Trudinger-Moser-Onofri inequality on $\mathbb{S}^2$ Optimal Aubin-Moser-Onofri inequality on $\mathbb{S}^2$ Bibliography
Abstract We prove a natural generalization of Kneser's oscillation and Hardy's inequality for Sturm‐Liouville differential expressions. In Particular, assuming − d/dxp 0 (x)+q 0 (x), x ∈ a, b), −∞≦a<b≦∞, …
Abstract We prove a natural generalization of Kneser's oscillation and Hardy's inequality for Sturm‐Liouville differential expressions. In Particular, assuming − d/dxp 0 (x)+q 0 (x), x ∈ a, b), −∞≦a<b≦∞, to be nonoscillatory near a (or b), we determine condition on q(x) such that − d/dxp 0 (x)+q 0 (x)+q(x) is nonoscillatory, respectively, oscillatory near a (or b)
Versions of Hardy's inequality involving radial derivatives and logarithmic refinements are deduced.
Versions of Hardy's inequality involving radial derivatives and logarithmic refinements are deduced.
Using the factorizations of suitable operators, we establish several identities that give simple and direct understandings as well as provide the remainders and “virtual” optimizers of several the Hardy and …
Using the factorizations of suitable operators, we establish several identities that give simple and direct understandings as well as provide the remainders and “virtual” optimizers of several the Hardy and Hardy–Rellich type inequalities.
In this paper, we study the Hardy–Rellich inequalities for polyharmonic operators in the critical dimension and an analogue in the p-biharmonic case. We also develop some optimal weighted Hardy–Sobolev inequalities …
In this paper, we study the Hardy–Rellich inequalities for polyharmonic operators in the critical dimension and an analogue in the p-biharmonic case. We also develop some optimal weighted Hardy–Sobolev inequalities in the general case and discuss the related eigenvalue problem. We also prove W 2,q (Ω) estimates in the biharmonic case.
We give sharp remainder terms of $L^{p}$ and weighted Hardy and Rellich inequalities on one of most general subclasses of nilpotent Lie groups, namely the class of homogeneous groups. As …
We give sharp remainder terms of $L^{p}$ and weighted Hardy and Rellich inequalities on one of most general subclasses of nilpotent Lie groups, namely the class of homogeneous groups. As consequences, we obtain analogues of the generalised classical Hardy and Rellich inequalities and the uncertainty principle on homogeneous groups. We also prove higher order inequalities of Hardy-Rellich type, all with sharp constants. A number of identities are derived including weighted and higher order types.
We show that the classical Hardy inequalities with optimal constants in the Sobolev spaces $W_0^{1,p}$ and in higher-order Sobolev spaces on a bounded domain $\Omega \subset \mathbb {R}^n$ can be …
We show that the classical Hardy inequalities with optimal constants in the Sobolev spaces $W_0^{1,p}$ and in higher-order Sobolev spaces on a bounded domain $\Omega \subset \mathbb {R}^n$ can be refined by adding remainder terms which involve $L^p$ norms. In the higher-order case further $L^p$ norms with lower-order singular weights arise. The case $1<p<2$ being more involved requires a different technique and is developed only in the space $W_0^{1,p}$.
Sharp $L^p$ extensions of Pitt's inequality expressed as a weighted Sobolev inequality are obtained using convolution estimates and Stein-Weiss potentials. Optimal constants are obtained for the full Stein-Weiss potential as …
Sharp $L^p$ extensions of Pitt's inequality expressed as a weighted Sobolev inequality are obtained using convolution estimates and Stein-Weiss potentials. Optimal constants are obtained for the full Stein-Weiss potential as a map from $L^p$ to itself which in turn yield semi-classical Rellich inequalities on $\mathbb {R}^n$. Additional results are obtained for Stein-Weiss potentials with gradient estimates and with mixed homogeneity. New proofs are given for the classical Pitt and Stein-Weiss estimates.
On etudie le caractere auto-adjoint des operateurs de Schrodinger sur Y=U j∈I (x j ,y j ) ou on suppose y j ≤x j+1
On etudie le caractere auto-adjoint des operateurs de Schrodinger sur Y=U j∈I (x j ,y j ) ou on suppose y j ≤x j+1