Strong maximum principle for Schrödinger operators with singular potential
Annales de l'I.H.P. Analyse non linéaire, Tome 33 (2016) no. 2, pp. 477-493.

We prove that for every p>1 and for every potential VLp, any nonnegative function satisfying Δu+Vu0 in an open connected set of RN is either identically zero or its level set {u=0} has zero W2,p capacity. This gives an affirmative answer to an open problem of Bénilan and Brezis concerning a bridge between Serrin–Stampacchia's strong maximum principle for p>N2 and Ancona's strong maximum principle for p=1. The proof is based on the construction of suitable test functions depending on the level set {u=0}, and on the existence of solutions of the Dirichlet problem for the Schrödinger operator with diffuse measure data.

DOI : 10.1016/j.anihpc.2014.11.004
Classification : 35B05, 35B50, 31B15, 31B35
Mots-clés : Maximum principle, Schrödinger operator, Kato's inequality, Capacity
@article{AIHPC_2016__33_2_477_0,
     author = {Orsina, Luigi and Ponce, Augusto C.},
     title = {Strong maximum principle for {Schr\"odinger} operators with singular potential},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {477--493},
     publisher = {Elsevier},
     volume = {33},
     number = {2},
     year = {2016},
     doi = {10.1016/j.anihpc.2014.11.004},
     zbl = {1342.35083},
     mrnumber = {3465383},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1016/j.anihpc.2014.11.004/}
}
TY  - JOUR
AU  - Orsina, Luigi
AU  - Ponce, Augusto C.
TI  - Strong maximum principle for Schrödinger operators with singular potential
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2016
SP  - 477
EP  - 493
VL  - 33
IS  - 2
PB  - Elsevier
UR  - http://archive.numdam.org/articles/10.1016/j.anihpc.2014.11.004/
DO  - 10.1016/j.anihpc.2014.11.004
LA  - en
ID  - AIHPC_2016__33_2_477_0
ER  - 
%0 Journal Article
%A Orsina, Luigi
%A Ponce, Augusto C.
%T Strong maximum principle for Schrödinger operators with singular potential
%J Annales de l'I.H.P. Analyse non linéaire
%D 2016
%P 477-493
%V 33
%N 2
%I Elsevier
%U http://archive.numdam.org/articles/10.1016/j.anihpc.2014.11.004/
%R 10.1016/j.anihpc.2014.11.004
%G en
%F AIHPC_2016__33_2_477_0
Orsina, Luigi; Ponce, Augusto C. Strong maximum principle for Schrödinger operators with singular potential. Annales de l'I.H.P. Analyse non linéaire, Tome 33 (2016) no. 2, pp. 477-493. doi : 10.1016/j.anihpc.2014.11.004. http://archive.numdam.org/articles/10.1016/j.anihpc.2014.11.004/

[1] Adams, David R.; Hedberg, Lars I. Function Spaces and Potential Theory, Grundlehren Math. Wiss., vol. 314, Springer-Verlag, Berlin, 1996 | DOI | MR | Zbl

[2] Ancona, Alano Une propriété d'invariance des ensembles absorbants par perturbation d'un opérateur elliptique, Commun. Partial Differ. Equ., Volume 4 (1979), pp. 321–337 | MR | Zbl

[3] Ancona, Alano Elliptic operators, conormal derivatives and positive parts of functions, J. Funct. Anal., Volume 257 (2009), pp. 2124–2158 (with an appendix by H. Brezis) | MR | Zbl

[4] Baras, Pierre; Pierre, Michel Singularités éliminables pour des équations semi-linéaires, Ann. Inst. Fourier (Grenoble), Volume 34 (1984), pp. 185–206 | Numdam | MR | Zbl

[5] Bénilan, Philippe; Brezis, Haïm Nonlinear problems related to the Thomas–Fermi equation, J. Evol. Equ., Volume 3 (2004), pp. 673–770 (dedicated to Ph. Bénilan) | MR | Zbl

[6] Boccardo, Lucio; Gallouët, Thierry; Orsina, Luigi Existence and uniqueness of entropy solutions for nonlinear elliptic equations with measure data, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 13 (1996), pp. 539–551 | Numdam | MR | Zbl

[7] Brelot, Marcel Sur l'allure des fonctions harmoniques et sousharmoniques à la frontière, Math. Nachr., Volume 4 (1951), pp. 298–307 | MR | Zbl

[8] Brezis, Haïm; Marcus, Moshe; Ponce, Augusto C. Nonlinear elliptic equations with measures revisited, Mathematical aspects of nonlinear dispersive equations, Ann. Math. Stud., vol. 163, Princeton University Press, Princeton, NJ, 2007, pp. 55–109 | MR | Zbl

[9] Brezis, Haïm; Ponce, Augusto C. Remarks on the strong maximum principle, Differ. Integral Equ., Volume 16 (2003), pp. 1–12 | MR | Zbl

[10] Brezis, Haïm; Ponce, Augusto C. Kato's inequality when Δu is a measure, C. R. Math. Acad. Sci. Paris, Volume 338 (2004), pp. 599–604 | MR | Zbl

[11] Brezis, Haïm; Ponce, Augusto C. Kato's inequality up to the boundary, Commun. Contemp. Math., Volume 10 (2008), pp. 1217–1241 | MR | Zbl

[12] Dal Maso, Gianni On the integral representation of certain local functionals, Ric. Mat., Volume 32 (1983), pp. 85–113 | MR | Zbl

[13] Dal Maso, Gianni; Murat, François; Orsina, Luigi; Prignet, Alain Renormalized solutions of elliptic equations with general measure data, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (4), Volume 28 (1999), pp. 741–808 | Numdam | MR | Zbl

[14] de la Vallée Poussin, Charles Extension de la méthode du balayage de Poincaré et problème de Dirichlet, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 2 (1932), pp. 169–232 | Numdam | MR | Zbl

[15] de la Vallée Poussin, Charles Potentiel et problème généralisé de Dirichlet, Math. Gaz., Volume 22 (1938), pp. 17–36 | JFM | Zbl

[16] Feyel, Denis; de la Pradelle, Arnaud Topologies fines et compactifications associées à certains espaces de Dirichlet, Ann. Inst. Fourier (Grenoble), Volume 27 (1977), pp. 121–146 | Numdam | MR | Zbl

[17] Gallouët, Thierry; Morel, Jean-Michel Resolution of a semilinear equation in L1 , Proc. R. Soc. Edinb. A, Volume 96 (1984), pp. 275–288 (Corrigendum) | MR | Zbl

[18] Gilbarg, David; Trudinger, Neil S. Elliptic Partial Differential Equations of Second Order, Grundlehren Math. Wiss., vol. 224, Springer-Verlag, Berlin, 1998 | MR | Zbl

[19] Kato, Tosio Schrödinger operators with singular potentials, Isr. J. Math., Volume 13 (1972), pp. 135–148 | MR | Zbl

[20] Littman, Walter; Stampacchia, Guido; Weinberger, Hans F. Regular points for elliptic equations with discontinuous coefficients, Ann. Sc. Norm. Super. Pisa (3), Volume 17 (1963), pp. 43–77 | Numdam | MR | Zbl

[21] Lucia, Marcello On the uniqueness and simplicity of the principal eigenvalue, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9), Mat. Appl., Volume 16 (2005), pp. 133–142 | MR | Zbl

[22] Maz'ya, Vladimir G. Problems of mathematical analysis, No. 3: Integral and differential operators, Differential equations, J. Sov. Math., Volume 1 (1972), pp. 33–68 (English transl.:, 1973, 205–234) | Zbl

[23] Maz'ya, Vladimir G.; Shaposhnikova, Tatyana O. Theory of Sobolev Multipliers, Grundlehren Math. Wiss., vol. 337, Springer-Verlag, Berlin, 2009 | MR | Zbl

[24] Montenegro, Marcelo; Ponce, Augusto C. The sub-supersolution method for weak solutions, Proc. Am. Math. Soc., Volume 136 (2008), pp. 2429–2438 | MR | Zbl

[25] Moser, Jürgen On Harnack's theorem for elliptic differential equations, Commun. Pure Appl. Math., Volume 14 (1961), pp. 577–591 | MR | Zbl

[26] Orsina, Luigi; Ponce, Augusto C. Semilinear elliptic equations and systems with diffuse measures, J. Evol. Equ., Volume 8 (2008), pp. 781–812 | MR | Zbl

[27] Ponce, Augusto C. Selected problems on elliptic equations involving measures, 2012 (winner monograph of the Concours annuel 2012 in Mathematics of the Académie royale de Belgique) | arXiv

[28] Ponce, Augusto C. Elliptic PDEs, Measures and Capacities. From the Poisson Equation to Nonlinear Thomas–Fermi Problems, EMS Tracts Math., vol. 23, European Mathematical Society (EMS), Zürich, 2015 (winner of the 2014 EMS Monograph Award) | MR | Zbl

[29] Schwartz, Laurent Généralisation de la notion de fonction, de dérivation, de transformation de Fourier et applications mathématiques et physiques, Ann. Univ. Grenoble Sect. Sci. Math. Phys. (N.S.), Volume 21 (1945), pp. 57–74 | Numdam | MR | Zbl

[30] Serrin, James Local behavior of solutions of quasi-linear equations, Acta Math., Volume 111 (1964), pp. 247–302 | MR | Zbl

[31] Stampacchia, Guido Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus, Ann. Inst. Fourier (Grenoble), Volume 15 (1965), pp. 189–258 | Numdam | MR | Zbl

[32] Stein, Elias M. Singular Integrals and Differentiability Properties of Functions, Princeton Math. Ser., vol. 30, Princeton University Press, Princeton, NJ, 1970 | MR | Zbl

[33] Trudinger, Neil S. Linear elliptic operators with measurable coefficients, Ann. Sc. Norm. Super. Pisa (3), Volume 27 (1973), pp. 265–308 | Numdam | MR | Zbl

[34] Van Schaftingen, Jean; Willem, Michel Symmetry of solutions of semilinear elliptic problems, J. Eur. Math. Soc., Volume 10 (2008), pp. 439–456 | MR | Zbl

Cité par Sources :