Positive Solutions to Schrödinger’s Equation and the Exponential Integrability of the Balayage

Frazier, Michael W.; Verbitsky, Igor E.

doi:10.5802/aif.3113

Positive Solutions to Schrödinger’s Equation and the Exponential Integrability of the Balayage
[Solutions positives de l’équation de Schrödinger et l’intégrabilité exponentielle du balayage.]

Frazier, Michael W.¹ ; Verbitsky, Igor E.²

¹ Department of Mathematics University of Tennessee Knoxville, Tennessee 37922 (USA)
² Department of Mathematics University of Missouri Columbia, Missouri 65211 (USA)

Annales de l'Institut Fourier, Tome 67 (2017) no. 4, pp. 1393-1425.

Résumé
Abstract

Soit $Ω \subset ℝ^{n} (n \geq 2)$ un domaine $C^{2}$ borné. Soit $q \in L_{l o c}^{1} (Ω)$ , avec $q \geq 0$ . 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 $(- Δ - q) u = 0$ , $u \geq 0$ sur $Ω$ et $u = 1$ sur $\partial Ω$ , et au problème non linéaire associé, avec une croissance quadratique par rapport au gradient, $- Δ u = {| \nabla u |}^{2} + q$ sur $Ω$ et $u = 0$ sur $\partial Ω$ . 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 $\partial Ω$ du balayage de la mesure $δ q d x$ , où $δ (x) = dist (x, \partial Ω)$ . Cette condition est optimale, et elle apparaît dans un tel contexte pour la première fois. Elle est notamment remplie si $δ q d x$ est une mesure de Carleson dans $Ω$ , ou si son balayage, de norme suffisament petite, est dans $BMO (\partial Ω)$ . Cela résout un problème qui était resté en suspens jusqu’à présent.

Let $Ω \subset ℝ^{n}$ , for $n \geq 2$ , be a bounded $C^{2}$ domain. Let $q \in L_{l o c}^{1} (Ω)$ with $q \geq 0$ . 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 $(- ▵ - q) u = 0$ , $u \geq 0$ on $Ω$ , $u = 1$ on $\partial Ω$ , and the related nonlinear problem with quadratic growth in the gradient, $- ▵ u = {| \nabla u |}^{2} + q$ on $Ω$ , $u = 0$ on $\partial Ω$ . 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 $\partial Ω$ of the balayage of the measure $δ q d x$ , where $δ (x) = dist (x, \partial Ω)$ . This condition is sharp, and appears in such a context for the first time. It holds, for example, if $δ q d x$ is a Carleson measure in $Ω$ , or if its balayage is in $B M O (\partial Ω)$ , with sufficiently small norm. This solves an open problem posed in the literature.

Reçu le : 2015-09-30
Révisé le : 2016-09-19
Accepté le : 2016-09-23
Publié le : 2017-09-26

DOI : 10.5802/aif.3113

Classification : 42B20, 60J65, 81Q15
Keywords: Schrödinger equation, very weak solutions, balayage, Carleson measures, BMO
Mot clés : Equation de Schrödinger, solutions très faibles, balayage, mesure de Carleson, BMO

Affiliations des auteurs :

Frazier, Michael W. ¹ ; Verbitsky, Igor E. ²

¹ Department of Mathematics University of Tennessee Knoxville, Tennessee 37922 (USA)
² Department of Mathematics University of Missouri Columbia, Missouri 65211 (USA)

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Positive {Solutions} to {Schr\"odinger{\textquoteright}s} {Equation} and the {Exponential} {Integrability} of the {Balayage}},
     journal = {Annales de l'Institut Fourier},
     pages = {1393--1425},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
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Frazier, Michael W.; Verbitsky, Igor E. Positive Solutions to Schrödinger’s Equation and the Exponential Integrability of the Balayage. Annales de l'Institut Fourier, Tome 67 (2017) no. 4, pp. 1393-1425. doi : 10.5802/aif.3113. https://aif.centre-mersenne.org/articles/10.5802/aif.3113/

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