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

Frazier, Michael W.; Verbitsky, Igor E.

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

Soit $\Omega \subset {ℝ}^n(n\ge 2)$ 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 $(-\Delta -q)u=0$, $u\ge 0$ sur $\Omega$ et $u=1$ sur $\partial \Omega$, et au problème non linéaire associé, avec une croissance quadratique par rapport au gradient, $-\Delta u={\left|\nabla u\right|}^2+q$ 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 $\delta q\mathrm{d}x$, où $\delta \left(x\right)=\mathrm{dist}(x,\partial \Omega )$. Cette condition est optimale, et elle apparaît dans un tel contexte pour la première fois. Elle est notamment remplie si $\delta q\mathrm{d}x$ 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.

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 $(-▵-q)u=0$, $u\ge 0$ on $\Omega$, $u=1$ on $\partial \Omega$, and the related nonlinear problem with quadratic growth in the gradient, $-▵u={\left|\nabla u\right|}^2+q$ 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 $\delta q\mathrm{d}x$, where $\delta \left(x\right)=\text{dist}(x,\partial \Omega )$. This condition is sharp, and appears in such a context for the first time. It holds, for example, if $\delta q\mathrm{d}x$ 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.

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

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