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
@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 = {https://aif.centre-mersenne.org/item/AIF_2017__67_4_1393_0}
}
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/

[1] Abdellaoui, Boumediene; Dall’Aglio, Andrea; Peral, Ireneo Some remarks on elliptic problems with critical growth in the gradient, J. Differ. Equations, Tome 222 (2006) no. 1, pp. 21-62 | Article | MR 2200746

[2] Abdellaoui, Boumediene; Dall’Aglio, Andrea; Peral, Ireneo Corrigendum to ‘Some remarks on elliptic problems with critical growth in the gradient’, J. Differ. Equations, Tome 246 (2009) no. 7, pp. 2988-2990 | Article | MR 2503032

[3] Adams, David R.; Hedberg, Lars Inge Function spaces and potential theory, Springer, Grundlehren der Mathematischen Wissenschaften, Tome 314 (1996), xii+366 pages | Article | MR 1411441

[4] Ancona, Alano First eigenvalues and comparison of Green’s functions for elliptic operators on manifolds or domains, J. Anal. Math., Tome 72 (1997), pp. 45-92 | Article | MR 1482989

[5] Armitage, David H.; Gardiner, Stephen J. Classical potential theory, Springer, Springer Monographs in Mathematics (2001), xvi+333 pages | Article | MR 1801253

[6] Brézis, Haim; Cazenave, Thierry; Martel, Yvan; Ramiandrisoa, Arthur Blow up for u t -Δu=g(u) revisited, Adv. Differ. Equ., Tome 1 (1996) no. 1, pp. 73-90 | MR 1357955

[7] Chung, Kai Lai; Zhao, Zhong Xin From Brownian motion to Schrödinger’s equation, Springer, Grundlehren der Mathematischen Wissenschaften, Tome 312 (1995), xii+287 pages | Article | MR 1329992

[8] Dávila, Juan; Dupaigne, Louis Comparison results for PDEs with a singular potential, Proc. R. Soc. Edinb., Sect. A, Tome 133 (2003) no. 1, pp. 61-83 | Article | MR 1960047

[9] Ferone, Vincenzo; Murat, François Quasilinear problems having quadratic growth in the gradient: an existence result when the source term is small, Équations aux dérivées partielles et applications, Gauthier-Villars (1998), pp. 497-515 | MR 1648236

[10] Ferone, Vincenzo; Murat, François Nonlinear problems having natural growth in the gradient: an existence result when the source terms are small, Nonlinear Anal., Tome 42 (2000) no. 7, pp. 1309-1326 | Article | MR 1780731

[11] Ferone, Vincenzo; Murat, François Nonlinear elliptic equations with natural growth in the gradient and source terms in Lorentz spaces, J. Differ. Equations, Tome 256 (2014) no. 2, pp. 577-608 | Article | MR 3121707

[12] Frazier, Michael; Nazarov, Fedor; Verbitsky, Igor E. Global estimates for kernels of Neumann series and Green’s functions, J. Lond. Math. Soc., Tome 90 (2014) no. 3, pp. 903-918 | Article | MR 3291806

[13] Frazier, Michael; Verbitsky, Igor E. Global Green’s function estimates, Around the research of Vladimir Mazʼya. III, Springer (International Mathematical Series (New York)) Tome 13 (2010), pp. 105-152 | Article | MR 2664707

[14] Garnett, John B. Bounded analytic functions, Springer, Graduate Texts in Mathematics, Tome 236 (2007), xiv+459 pages | MR 2261424

[15] GrigorʼYan, Alexander; Verbitsky, Igor E. Pointwise estimates of solutions to semilinear elliptic equations and inequalities (https://arxiv.org/abs/1511.03188, to appear in J. Anal. Math.)

[16] Hamid, Haydar Abdel; Bidaut-Veron, Marie Françoise On the connection between two quasilinear elliptic problems with source terms of order 0 or 1, Commun. Contemp. Math., Tome 12 (2010) no. 5, pp. 727-788 | Article | MR 2733197

[17] Hansen, Wolfhard Global comparison of perturbed Green functions, Math. Ann., Tome 334 (2006) no. 3, pp. 643-678 | Article | MR 2207878

[18] Hansen, Wolfhard; Netuka, Ivan On the Picard principle for Δ+μ, Math. Z., Tome 270 (2012) no. 3-4, pp. 783-807 | Article | MR 2892924

[19] Hansson, Kurt; MazʼYa, Vladimir G.; Verbitsky, Igor E. Criteria of solvability for multidimensional Riccati equations, Ark. Mat., Tome 37 (1999) no. 1, pp. 87-120 | Article | MR 1673427

[20] Heinonen, Juha; Kilpeläinen, Tero; Martio, Olli Nonlinear potential theory of degenerate elliptic equations, Dover Publications, Inc., Mineola, NY (2006), xii+404 pages (Unabridged republication of the 1993 original) | MR 2305115

[21] Jaye, Benjamin J.; MazʼYa, Vladimir G.; Verbitsky, Igor E. Existence and regularity of positive solutions of elliptic equations of Schrödinger type, J. Anal. Math., Tome 118 (2012) no. 2, pp. 577-621 | Article | MR 3000692

[22] Landkof, N. S. Foundations of modern potential theory, Springer (1972), x+424 pages (Translated from the Russian by A. P. Doohovskoy, Die Grundlehren der mathematischen Wissenschaften, Band 180) | MR 0350027

[23] Malý, Jan; Ziemer, William P. Fine regularity of solutions of elliptic partial differential equations, American Mathematical Society, Mathematical Surveys and Monographs, Tome 51 (1997), xiv+291 pages | Article | MR 1461542

[24] Marcus, Moshe; Véron, Laurent Nonlinear second order elliptic equations involving measures, De Gruyter, Berlin, De Gruyter Series in Nonlinear Analysis and Applications, Tome 21 (2014), xiii+248 pages | MR 3156649

[25] MazʼYa, Vladimir G. Sobolev spaces with applications to elliptic partial differential equations, Springer, Grundlehren der Mathematischen Wissenschaften, Tome 342 (2011), xxviii+866 pages | Article | MR 2777530

[26] Murata, Minoru Structure of positive solutions to (-Δ+V)u=0 in R n , Duke Math. J., Tome 53 (1986) no. 4, pp. 869-943 | Article | MR 874676

[27] Pinchover, Yehuda Maximum and anti-maximum principles and eigenfunctions estimates via perturbation theory of positive solutions of elliptic equations, Math. Ann., Tome 314 (1999) no. 3, pp. 555-590 | Article | MR 1704549

[28] Pott, Sandra; Volberg, Alexander Carleson measure and balayage, Int. Math. Res. Not. (2010) no. 13, pp. 2427-2436 | Article | MR 2669654

[29] Widman, Kjell-Ove Inequalities for the Green function and boundary continuity of the gradient of solutions of elliptic differential equations, Math. Scand., Tome 21 (1967), p. 17-37 (1968) | Article | MR 0239264

[30] Zhao, Zhong Xin Green function for Schrödinger operator and conditioned Feynman-Kac gauge, J. Math. Anal. Appl., Tome 116 (1986) no. 2, pp. 309-334 | Article | MR 842803