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 Ω n , for n2, be a bounded C 2 domain. Let qL loc 1 (Ω) with q0. 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, u0 on Ω, u=1 on Ω, and the related nonlinear problem with quadratic growth in the gradient, -u=|u| 2 +q on Ω, u=0 on 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 Ω of the balayage of the measure δqdx, where δ(x)=dist(x,Ω). This condition is sharp, and appears in such a context for the first time. It holds, for example, if δqdx is a Carleson measure in Ω, or if its balayage is in BMO(Ω), with sufficiently small norm. This solves an open problem posed in the literature.

Soit Ω n (n2) un domaine C 2 borné. Soit qL loc 1 (Ω), avec q0. 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, u0 sur Ω et u=1 sur Ω, et au problème non linéaire associé, avec une croissance quadratique par rapport au gradient, -Δu=|u| 2 +q sur Ω et u=0 sur 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 Ω du balayage de la mesure δqdx, où δ(x)=dist(x,Ω). Cette condition est optimale, et elle apparaît dans un tel contexte pour la première fois. Elle est notamment remplie si δqdx est une mesure de Carleson dans Ω, ou si son balayage, de norme suffisament petite, est dans BMO(Ω). Cela résout un problème qui était resté en suspens jusqu’à présent.

Received : 2015-09-30
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
     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 = {http://www.numdam.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. http://www.numdam.org/item/AIF_2017__67_4_1393_0/

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