Hardy-type inequalities related to degenerate elliptic differential operators

Lorenzo D’Ambrosio

Type: Article

Publication Date: 2009-12-04

Citations: 82

DOI: https://doi.org/10.2422/2036-2145.2005.3.04

Abstract

We prove some Hardy-type inequalities related to quasilinear secondorder degenerate elliptic differential operatorsholds.We find an explicit value of the constant involved, which, in most cases, results optimal.As particular case we derive Hardy inequalities for subelliptic operators on Carnot Groups.Mathematics Subject Classification (2000): 35H10 (primary); 22E30, 26D10, 46E35 (secondary).Our principal result can be roughly described as follows: if φ : → R is any positive weight, for any u ∈ C 1 0 ( ) we have c |u| p φ p |∇ L φ| p dξ ≤ |∇ L u| p dξ, provided -L p φ ≥ 0.

Locations

ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE - View - PDF

Similar Works

Action	Title	Year	Authors
+	Hardy Type Inequalities Related to Degenerate Elliptic Differential Operators	2006	Lorenzo D’Ambrosio
+	Hardy Inequalities for a Class of Degenerate Elliptic Operators	2014	Stefanie Sonner
+	On families of optimal Hardy-weights for linear second-order elliptic operators	2019	Yehuda Pinchover Idan Versano
+	On families of optimal Hardy-weights for linear second-order elliptic operators	2019	Yehuda Pinchover Idan Versano
+	On families of optimal Hardy-weights for linear second-order elliptic operators	2019	Yehuda Pinchover Idan Versano
+	Hardy-type inequalities and Pohozaev-type identities for a class of -degenerate subelliptic operators and applications	2003	Huiqing Zhang Pengcheng Niu
+ PDF Chat	Kato's Inequalities for Degenerate Quasilinear Elliptic Operators	2008	Toshio Horiuchi
+	Hardy inequalities on Riemannian manifolds and applications	2012	Lorenzo D’Ambrosio Serena Dipierro
+	Hardy inequalities on Riemannian manifolds and applications	2012	Lorenzo D’Ambrosio Serena Dipierro
+	Uniform Sobolev inequalities for second order non-elliptic differential operators	2015	Eun-Hee Jeong Yehyun Kwon Sanghyuk Lee
+ PDF Chat	Weighted Sobolev Spaces and Degenerate Elliptic Equations	2008	Albo Carlos Cavalheiro
+	Two-weight Sobolev-Poincaré inequalities and Harnack inequality for a class of degenerate elliptic operators	1994	Bruno Franchi Cristian E. Gutiérrez Richard L. Wheeden
+	Hardy Spaces for Second-Order Weakly Elliptic Operators in the Complex Plane	2023	Dorina Mitrea Irina Mitrea Marius Mitrea
+ PDF Chat	Multiplicity results for a degenerate quasilinear elliptic equation in half-space	2009	Ronaldo B. Assunção Paulo César Carrião Olı́mpio H. Miyagaki
+ PDF Chat	Nonnegative solutions of some quasilinear elliptic inequalities and applications	2010	Lorenzo D’Ambrosio Enzo Mitidieri
+	Hölder Estimates for A Class of Degenerate Elliptic Equations	2013	Qiaozhen Song Lihe Wang Dongsheng Li
+	Weighted Sobolev-Poincare Inequalities and Pointwise Estimates for a Class of Degenerate Elliptic Equations	1991	Bruno Franchi
+	Quasiconformal mappings and degenerate elliptic and parabolic equations	1987	Filippo Chiarenza Michelangelo Franciosi
+	First order Hardy inequalities revisited	2021	Xia Huang Dong Ye
+	Hlder continuity of weak solutions for degenerate quasilinear elliptic equations with different weights	2010	Xue Cui

Works That Cite This (78)

Action	Title	Year	Authors
+	Coercive Inequalities in Higher-Dimensional Anisotropic Heisenberg Group	2021	Esther Bou Dagher Bogusław Zegarliński
+	Optimal Hardy inequalities for general elliptic operators with improvements	2010	Craig Cowan
+	An Extension Problem and Trace Hardy Inequality for the Sublaplacian on H-Type Groups	2020	Luz Roncal Sundaram Thangavelu
+ PDF Chat	Reverse Stein–Weiss, Hardy–Littlewood–Sobolev, Hardy, Sobolev and Caffarelli–Kohn–Nirenberg inequalities on homogeneous groups	2022	Aidyn Kassymov Michael Ruzhansky Durvudkhan Suragan
+ PDF Chat	Improvements and Generalizations of Two Hardy Type Inequalities and Their Applications to the Rellich Type Inequalities	2022	Megumi Sano
+ PDF Chat	Horizontal magnetic fields and improved Hardy inequalities in the Heisenberg group	2023	Biagio Cassano Valentina Franceschi David Krejčiřı́k Dario Prandi
+ PDF Chat	Caccioppoli-type estimates and Hardy-type inequalities derived from weighted p-harmonic problems	2019	Iwona Chlebicka Pavel Drábek Agnieszka Kałamajska
+ PDF Chat	Critical growth problems with singular nonlinearities on Carnot groups	2015	Annunziata Loiudice
+ PDF Chat	Geometric inequalities for fractional Laplace operators and applications	2015	Eleonora Cinti Fausto Ferrari
+ PDF Chat	Hardy Inequalities with Best Constants on Finsler Metric Measure Manifolds	2019	Wei Zhao

Works Cited by This (49)

Action	Title	Year	Authors
+	Representation formulae and inequalities for solutions of a class of second order partial differential equations	2005	Lorenzo D’Ambrosio Enzo Mitidieri Stanislav I. Pohožaev
+	Symmetry properties of positive entire solutions of Yamabe-type equations on groups of Heisenberg type	2001	Nicola Garofalo Dimiter Vassilev
+	Hardy spaces on homogeneous groups	1982	Gerald B. Folland Elias M. Stein
+ PDF Chat	Hardy type and Rellich type inequalities on the Heisenberg group	2001	Pengcheng Niu Huiqing Zhang Yong Wang
+ PDF Chat	Hardy inequalities with optimal constants and remainder terms	2003	Filippo Gazzola Hans-Christoph Grunau Enzo Mitidieri
+ PDF Chat	On the best constant for Hardy’s inequality in $\mathbb {R}^n$	1998	Moshe Marcus Victor J. Mizel Yehuda Pinchover
+	Hardy inequalities related to Grushin type operators	2003	Lorenzo D’Ambrosio
+	The 𝑃-Laplace equation on a class of Grushin-type spaces	2006	Thomas Bieske Jasun Gong
+	Nonlinear Liouville theorems for some critical problems on H-type groups	2003	Andrea Bonfiglioli Francesco Uguzzoni
+	Nonlinear Liouville theorems for Grushin and Tricomi operators	2003	Lorenzo D’Ambrosio Sandra Lucente