Strong maximum principle for Schrödinger operators with singular potential

Orsina, Luigi; Ponce, Augusto C.

doi:10.1016/j.anihpc.2014.11.004

Orsina, Luigi ; Ponce, Augusto C.

Annales de l'I.H.P. Analyse non linéaire, Tome 33 (2016) no. 2, pp. 477-493

Cet article a éte moissonné depuis la source Numdam

Voir la notice de l'article

Résumé

We prove that for every $p > 1$ and for every potential $V \in L^{p}$ , any nonnegative function satisfying $- Δ u + V u \geq 0$ in an open connected set of $R^{N}$ is either identically zero or its level set ${u = 0}$ has zero $W^{2, p}$ 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 $p > \frac{N}{2}$ 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.

MR Zbl

DOI : 10.1016/j.anihpc.2014.11.004

Classification : 35B05, 35B50, 31B15, 31B35
Keywords: 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},
     year = {2016},
     publisher = {Elsevier},
     volume = {33},
     number = {2},
     doi = {10.1016/j.anihpc.2014.11.004},
     zbl = {1342.35083},
     mrnumber = {3465383},
     language = {en},
     url = {http://geodesic.mathdoc.fr/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://geodesic.mathdoc.fr/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://geodesic.mathdoc.fr/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

Cité par Sources :

Parcourir par

Geodesic

Parcourir par