Kato's inequality when Δ<i>u</i> is a measure

Haı̈m Brezis; Augusto C. Ponce

doi:10.1016/j.crma.2003.12.032

Partial Differential Equations

Kato's inequality when Δu is a measure
[L'inégalité de Kato lorsque Δu est une mesure]

Présenté par : Haïm Brezis

Haı̈m Brezis ^1,² ; Augusto C. Ponce ^1,²

¹ Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, BC 187, 4, pl. Jussieu, 75252 Paris cedex 05, France
² Rutgers University, Department of Math., Hill Center, Busch Campus, 110 Frelinghuysen Rd, Piscataway, NJ 08854, USA

Comptes Rendus. Mathématique, Volume 338 (2004) no. 8, pp. 599-604.

Abstract (VO)
Résumé

We extend the classical version of Kato's inequality in order to allow functions u∈L¹_loc such that Δu is a Radon measure. This inequality has been recently applied by Brezis, Marcus, and Ponce to study the existence of solutions of the nonlinear equation −Δu+g(u)=μ, where μ is a measure and $g : ℝ \to ℝ$ is a nondecreasing continuous function.

Nous étendons l'inégalité de Kato classique à des fonctions u∈L¹_loc telles que Δu est une mesure de Radon. Cette inégalité a été récemment utilisée par Brezis, Marcus et Ponce pour étudier l'existence de solutions de l'équation elliptique non linéaire −Δu+g(u)=μ, où μ est une mesure et $g : ℝ \to ℝ$ est une fonction croissante et continue.

Reçu le : 2003-12-17
Accepté le : 2003-12-17
Publié le : 2004-03-25

DOI : 10.1016/j.crma.2003.12.032

Affiliations des auteurs :

Haı̈m Brezis ^{1, 2} ; Augusto C. Ponce ^{1, 2}

¹ Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, BC 187, 4, pl. Jussieu, 75252 Paris cedex 05, France
² Rutgers University, Department of Math., Hill Center, Busch Campus, 110 Frelinghuysen Rd, Piscataway, NJ 08854, USA

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     title = {Kato's inequality when {\ensuremath{\Delta}\protect\emph{u}} is a measure},
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     pages = {599--604},
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Haı̈m Brezis; Augusto C. Ponce. Kato's inequality when Δu is a measure. Comptes Rendus. Mathématique, Volume 338 (2004) no. 8, pp. 599-604. doi : 10.1016/j.crma.2003.12.032. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2003.12.032/

Bibliographie
Cité par

[1] A. Ancona Une propriété d'invariance des ensembles absorbants par perturbation d'un opérateur elliptique, Comm. Partial Differential Equations, Volume 4 (1979), pp. 321-337

[2] L. Boccardo; T. Gallouët; L. Orsina Existence and uniqueness of entropy solutions for nonlinear elliptic equations with measure data, Ann. Inst. H. Poincaré Anal. Non Linéaire, Volume 13 (1996), pp. 539-551

[3] H. Brezis; T. Cazenave; Y. Martel; A. Ramiandrisoa Blow up for u_t−Δu=g(u) revisited, Adv. Differential Equations, Volume 1 (1996), pp. 73-90

[4] H. Brezis; A.C. Ponce Remarks on the strong maximum principle, Differential Integral Equations, Volume 16 (2003), pp. 1-12

[5] H. Brezis, M. Marcus, A.C. Ponce, Nonlinear elliptic equations with measures revisited, in preparation

[6] L. Dupaigne, A.C. Ponce, Singularities of positive supersolutions in elliptic PDEs, Selecta Math. (N.S.), in press

[7] M. Fukushima; K. Sato; S. Taniguchi On the closable part of pre-Dirichlet forms and the fine supports of underlying measures, Osaka Math. J., Volume 28 (1991), pp. 517-535

[8] T. Kato Schrödinger operators with singular potentials, Israel J. Math., Volume 13 (1972), pp. 135-148

Juan Huang; Sheng Wang Normalized ground states for the mass supercritical Schrödinger-Bopp-Podolsky system: Existence, uniqueness, limit behavior, strong instability, Journal of Differential Equations, Volume 437 (2025), p. 113282 | DOI:10.1016/j.jde.2025.113282
Francescantonio Oliva; Francesco Petitta Singular elliptic PDEs: an extensive overview, Partial Differential Equations and Applications, Volume 6 (2025) no. 1 | DOI:10.1007/s42985-024-00308-9
Stephen J. Gardiner; Tomas Sjödin Partial Balayage for the Helmholtz Equation, Potential Analysis (2025) | DOI:10.1007/s11118-025-10217-0
Alon Nishry; Aron Wennman The forbidden region for random zeros: Appearance of quadrature domains, Communications on Pure and Applied Mathematics, Volume 77 (2024) no. 3, p. 1766 | DOI:10.1002/cpa.22142
Mousomi Bhakta; Moshe Marcus; Phuoc-Tai Nguyen Boundary value problems for semilinear Schrödinger equations with singular potentials and measure data, Mathematische Annalen, Volume 390 (2024) no. 1, p. 351 | DOI:10.1007/s00208-023-02764-x
Batu Güneysu; Stefano Pigola; Peter Stollmann; Giona Veronelli A new notion of subharmonicity on locally smoothing spaces, and a conjecture by Braverman, Milatovic, Shubin, Mathematische Annalen, Volume 390 (2024) no. 3, p. 4209 | DOI:10.1007/s00208-024-02855-3
Pu-Zhao Kow; Henrik Shahgholian Multi-phase k-quadrature domains and applications to acoustic waves and magnetic fields, Partial Differential Equations and Applications, Volume 5 (2024) no. 3 | DOI:10.1007/s42985-024-00283-1
Emeric Bouin; Jean Dolbeault; Laurent Lafleche Fractional Hypocoercivity, Communications in Mathematical Physics, Volume 390 (2022) no. 3, p. 1369 | DOI:10.1007/s00220-021-04296-4
Émeric Bouin; Clément Mouhot Quantitative fluid approximation in transport theory: a unified approach, Probability and Mathematical Physics, Volume 3 (2022) no. 3, p. 491 | DOI:10.2140/pmp.2022.3.491
Jean Michel Rakotoson A metric potential capacity: some qualitative properties of Schrödinger’s equations with a non negative potential, Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, Volume 116 (2022) no. 3 | DOI:10.1007/s13398-022-01254-0
Stefano Buccheri; Luigi Orsina; Augusto C. Ponce An Agmon–Allegretto–Piepenbrink principle for Schrödinger operators, Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, Volume 116 (2022) no. 4 | DOI:10.1007/s13398-022-01293-7
Luigi Orsina; Augusto C. Ponce On the nonexistence of Green's function and failure of the strong maximum principle, Journal de Mathématiques Pures et Appliquées, Volume 134 (2020), p. 72 | DOI:10.1016/j.matpur.2019.06.001
Grzegorz Karch; Moritz Kassmann; Miłosz Krupski A Framework for Nonlocal, Nonlinear Initial Value Problems, SIAM Journal on Mathematical Analysis, Volume 52 (2020) no. 3, p. 2383 | DOI:10.1137/19m124143x
Jean Michel Rakotoson Potential-capacity and some applications, Asymptotic Analysis, Volume 114 (2019) no. 3-4, p. 225 | DOI:10.3233/asy-191523
Xiaojing Liu; Toshio Horiuchi Kato's inequalities for admissible functions to quasilinear elliptic operators A , Mathematical Journal of Ibaraki University, Volume 51 (2019) no. 0, p. 49 | DOI:10.5036/mjiu.51.49
Björn Gustafsson; Joakim Roos Partial balayage on Riemannian manifolds, Journal de Mathématiques Pures et Appliquées, Volume 118 (2018), p. 82 | DOI:10.1016/j.matpur.2017.07.013
Francescantonio Oliva; Francesco Petitta Finite and infinite energy solutions of singular elliptic problems: Existence and uniqueness, Journal of Differential Equations, Volume 264 (2018) no. 1, p. 311 | DOI:10.1016/j.jde.2017.09.008
Xiaojing Liu; Toshio Horiuchi The equivalences among p-capacity, p-Laplace-capacities and Hausdorff measure, Mathematical Journal of Ibaraki University, Volume 50 (2018) no. 0, p. 5 | DOI:10.5036/mjiu.50.5
Björn Gustafsson; Mihai Putinar Introduction, Hyponormal Quantization of Planar Domains, Volume 2199 (2017), p. 1 | DOI:10.1007/978-3-319-65810-0_1
Björn Gustafsson; Mihai Putinar Comparison with Classical Function Spaces, Hyponormal Quantization of Planar Domains, Volume 2199 (2017), p. 119 | DOI:10.1007/978-3-319-65810-0_8
Björn Gustafsson; Mihai Putinar Hilbert Space Factorization, Hyponormal Quantization of Planar Domains, Volume 2199 (2017), p. 23 | DOI:10.1007/978-3-319-65810-0_3
Björn Gustafsson; Mihai Putinar Exponential Orthogonal Polynomials, Hyponormal Quantization of Planar Domains, Volume 2199 (2017), p. 47 | DOI:10.1007/978-3-319-65810-0_4
Björn Gustafsson; Mihai Putinar Finite Central Truncations of Linear Operators, Hyponormal Quantization of Planar Domains, Volume 2199 (2017), p. 57 | DOI:10.1007/978-3-319-65810-0_5
Björn Gustafsson; Mihai Putinar The Exponential Transform, Hyponormal Quantization of Planar Domains, Volume 2199 (2017), p. 7 | DOI:10.1007/978-3-319-65810-0_2
Björn Gustafsson; Mihai Putinar Mother Bodies, Hyponormal Quantization of Planar Domains, Volume 2199 (2017), p. 77 | DOI:10.1007/978-3-319-65810-0_6
Björn Gustafsson; Mihai Putinar Examples, Hyponormal Quantization of Planar Domains, Volume 2199 (2017), p. 93 | DOI:10.1007/978-3-319-65810-0_7
Luigi Orsina; Augusto C. Ponce Strong maximum principle for Schrödinger operators with singular potential, Annales de l'Institut Henri Poincaré C, Analyse non linéaire, Volume 33 (2016) no. 2, p. 477 | DOI:10.1016/j.anihpc.2014.11.004
Tomasz Klimsiak Reduced measures for semilinear elliptic equations involving Dirichlet operators, Calculus of Variations and Partial Differential Equations, Volume 55 (2016) no. 4 | DOI:10.1007/s00526-016-1023-6
Hui-Chun Zhang; Xi-Ping Zhu Local Li–Yau’s estimates on RCD $^{*} (K, N)$ ∗ ( K , N ) metric measure spaces, Calculus of Variations and Partial Differential Equations, Volume 55 (2016) no. 4 | DOI:10.1007/s00526-016-1040-5
Mayte Pérez-Llanos Optimal power for an elliptic equation related to some Caffarelli-Kohn-Nirenberg inequalities, Communications on Pure and Applied Analysis, Volume 15 (2016) no. 6, p. 1975 | DOI:10.3934/cpaa.2016024
Xiaojing Liu; Toshio Horiuchi Remarks on Kato's inequality when ∆_pu is a measure , Mathematical journal of Ibaraki University, Volume 48 (2016) no. 0, p. 45 | DOI:10.5036/mjiu.48.45
Stephen J. Gardiner; Tomas Sjödin Stationary Boundary Points for a Laplacian Growth Problem in Higher Dimensions, Archive for Rational Mechanics and Analysis, Volume 213 (2014) no. 2, p. 503 | DOI:10.1007/s00205-014-0750-0
Boumediene Abdellaoui; Daniela Giachetti; Ireneo Peral; Magdalena Walias Elliptic problems with nonlinear terms depending on the gradient and singular on the boundary: Interaction with a Hardy-Leray potential, Discrete Continuous Dynamical Systems - A, Volume 34 (2014) no. 5, p. 1747 | DOI:10.3934/dcds.2014.34.1747
Stephen J. Gardiner; Tomas Sjödin Quadrature Domains and Their Two-Phase Counterparts, Harmonic and Complex Analysis and its Applications (2014), p. 261 | DOI:10.1007/978-3-319-01806-5_5
Stephen J. Gardiner; Tomas Sjödin Two-phase quadrature domains, Journal d'Analyse Mathématique, Volume 116 (2012) no. 1, p. 335 | DOI:10.1007/s11854-012-0009-3
Moshe Marcus; Augusto C. Ponce Reduced limits for nonlinear equations with measures, Journal of Functional Analysis, Volume 258 (2010) no. 7, p. 2316 | DOI:10.1016/j.jfa.2009.09.007
Alano Ancona Elliptic operators, conormal derivatives and positive parts of functions (with an appendix by Haïm Brezis), Journal of Functional Analysis, Volume 257 (2009) no. 7, p. 2124 | DOI:10.1016/j.jfa.2008.12.019
Boumediene Abdellaoui; Ireneo Peral; Ana Primo Breaking of resonance and regularizing effect of a first order quasi-linear term in some elliptic equations, Annales de l'Institut Henri Poincaré C, Analyse non linéaire, Volume 25 (2008) no. 5, p. 969 | DOI:10.1016/j.anihpc.2007.06.003
Francesco Petitta Renormalized solutions of nonlinear parabolic equations with general measure data, Annali di Matematica Pura ed Applicata, Volume 187 (2008) no. 4, p. 563 | DOI:10.1007/s10231-007-0057-y
HAÏM BREZIS; AUGUSTO C. PONCE KATO'S INEQUALITY UP TO THE BOUNDARY, Communications in Contemporary Mathematics, Volume 10 (2008) no. 06, p. 1217 | DOI:10.1142/s0219199708003241
Louis Dupaigne; Augusto C. Ponce; Alessio Porretta Elliptic equations with vertical asymptotes in the nonlinear term, Journal d'Analyse Mathématique, Volume 98 (2006) no. 1, p. 349 | DOI:10.1007/bf02790280
Haı¨m Brezis; Augusto C. Ponce Reduced measures on the boundary, Journal of Functional Analysis, Volume 229 (2005) no. 1, p. 95 | DOI:10.1016/j.jfa.2004.12.001

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