# ANNALES DE L'INSTITUT FOURIER

Positive Solutions to Schrödinger’s Equation and the Exponential Integrability of the Balayage
Annales de l'Institut Fourier, Volume 67 (2017) no. 4, p. 1393-1425

Let $\Omega \subset {ℝ}^{n}$, for $n\ge 2$, be a bounded ${C}^{2}$ domain. Let $q\in {L}_{loc}^{1}\left(\Omega \right)$ with $q\ge 0$. We give necessary conditions and matching sufficient conditions, which differ only in the constants involved, for the existence of very weak solutions to the boundary value problem $\left(-▵-q\right)u=0$, $u\ge 0$ on $\Omega$, $u=1$ on $\partial \Omega$, and the related nonlinear problem with quadratic growth in the gradient, $-▵u={|\nabla u|}^{2}+q$ on $\Omega$, $u=0$ on $\partial \Omega$. We also obtain precise pointwise estimates of solutions up to the boundary.

A crucial role is played by a new “boundary condition” on $q$ which is expressed in terms of the exponential integrability on $\partial \Omega$ of the balayage of the measure $\delta q\phantom{\rule{0.166667em}{0ex}}\mathrm{d}x$, where $\delta \left(x\right)=\text{dist}\left(x,\partial \Omega \right)$. This condition is sharp, and appears in such a context for the first time. It holds, for example, if $\delta q\phantom{\rule{0.166667em}{0ex}}\mathrm{d}x$ is a Carleson measure in $\Omega$, or if its balayage is in $BMO\left(\partial \Omega \right)$, with sufficiently small norm. This solves an open problem posed in the literature.

Soit $\Omega \subset {ℝ}^{n}\phantom{\rule{0.277778em}{0ex}}\left(n\ge 2\right)$ un domaine ${C}^{2}$ borné. Soit $q\in {L}_{\mathrm{l}oc}^{1}\left(\Omega \right)$, avec $q\ge 0$. Nous obtenons des conditions nécessaires et des conditions suffisantes correspondantes — dont seules les constantes impliquées diffèrent — pour l’éxistence de solutions très faibles au problème aux limites $\left(-\Delta -q\right)u=0$, $u\ge 0$ sur $\Omega$ et $u=1$ sur $\partial \Omega$, et au problème non linéaire associé, avec une croissance quadratique par rapport au gradient, $-\Delta u={|\nabla u|}^{2}+q$ sur $\Omega$ et $u=0$ sur $\partial \Omega$. Nous parvenons aussi à des estimations ponctuelles précises des solutions jusqu’à la frontière.

Un rôle crucial est joué par une nouvelle “condition aux limites” portant sur $q$, exprimée en terme d’intégrabilité exponentielle sur $\partial \Omega$ du balayage de la mesure $\delta q\phantom{\rule{0.166667em}{0ex}}\mathrm{d}x$, où $\delta \left(x\right)=\mathrm{dist}\left(x,\partial \Omega \right)$. Cette condition est optimale, et elle apparaît dans un tel contexte pour la première fois. Elle est notamment remplie si $\delta q\phantom{\rule{0.166667em}{0ex}}\mathrm{d}x$ est une mesure de Carleson dans $\Omega$, ou si son balayage, de norme suffisament petite, est dans $\mathrm{BMO}\left(\partial \Omega \right)$. Cela résout un problème qui était resté en suspens jusqu’à présent.

Revised : 2016-09-19
Accepted : 2016-09-23
Published online : 2017-09-26
DOI : https://doi.org/10.5802/aif.3113
Classification:  42B20,  60J65,  81Q15
Keywords: Schrödinger equation, very weak solutions, balayage, Carleson measures, BMO
@article{AIF_2017__67_4_1393_0,
author = {Frazier, Michael W. and Verbitsky, Igor E.},
title = {Positive Solutions to Schr\"odinger's Equation and the Exponential Integrability of the Balayage},
journal = {Annales de l'Institut Fourier},
publisher = {Association des Annales de l'institut Fourier},
volume = {67},
number = {4},
year = {2017},
pages = {1393-1425},
doi = {10.5802/aif.3113},
language = {en},
url = {aif.centre-mersenne.org/item/AIF_2017__67_4_1393_0}
}

Frazier, Michael W.; Verbitsky, Igor E. Positive Solutions to Schrödinger’s Equation and the Exponential Integrability of the Balayage. Annales de l'Institut Fourier, Volume 67 (2017) no. 4, pp. 1393-1425. doi : 10.5802/aif.3113. https://aif.centre-mersenne.org/item/AIF_2017__67_4_1393_0/

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