×

Regularity of solutions to a fractional elliptic problem with mixed Dirichlet-Neumann boundary data. (English) Zbl 1476.35075

Summary: In this work we study regularity properties of solutions to fractional elliptic problems with mixed Dirichlet-Neumann boundary data when dealing with the spectral fractional Laplacian.

MSC:

35B65 Smoothness and regularity of solutions to PDEs
35J25 Boundary value problems for second-order elliptic equations
35R11 Fractional partial differential equations
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] B. Abdellaoui, A. Dieb and E. Valdinoci, A nonlocal concave-convex problem with nonlocal mixed boundary data, Commun. Pure Appl. Anal. 17 (2018), no. 3, 1103-1120. · Zbl 1395.35190
[2] B. Barrios and M. Medina, Strong maximum principles for fractional elliptic and parabolic problems with mixed boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A, doi:10.1017/prm.2018.77. · Zbl 1436.35059
[3] C. Brändle, E. Colorado, A. de Pablo and U. Sánchez, A concave-convex elliptic problem involving the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A 143 (2013), no. 1, 39-71. · Zbl 1290.35304
[4] X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian, Adv. Math. 224 (2010), no. 5, 2052-2093. · Zbl 1198.35286
[5] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations 32 (2007), no. 7-9, 1245-1260. · Zbl 1143.26002
[6] J. Carmona, E. Colorado, T. Leonori and A. Ortega, Semilinear fractional elliptic problems with mixed Dirichlet-Neumann boundary conditions, preprint (2019), https://arxiv.org/abs/1902.08925v1.
[7] E. Colorado and A. Ortega, The Brezis-Nirenberg problem for the fractional Laplacian with mixed Dirichlet-Neumann boundary conditions, J. Math. Anal. Appl. 473 (2019), no. 2, 1002-1025. · Zbl 1516.35455
[8] E. Colorado and I. Peral, Semilinear elliptic problems with mixed Dirichlet-Neumann boundary conditions, J. Funct. Anal. 199 (2003), no. 2, 468-507. · Zbl 1034.35041
[9] S. Dipierro, X. Ros-Oton and E. Valdinoci, Nonlocal problems with Neumann boundary conditions, Rev. Mat. Iberoam. 33 (2017), no. 2, 377-416. · Zbl 1371.35322
[10] E. B. Fabes, C. E. Kenig and R. P. Serapioni, The local regularity of solutions of degenerate elliptic equations, Comm. Partial Differential Equations 7 (1982), no. 1, 77-116. · Zbl 0498.35042
[11] D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Pure Appl. Math. 88, Academic Press, New York, 1980. · Zbl 0457.35001
[12] J.-L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications. Vol. I, Grundlehren Math. Wiss. 181, Springer, New York, 1972, · Zbl 0223.35039
[13] G. Molica Bisci, V. D. Radulescu and R. Servadei, Variational Methods for Nonlocal Fractional Problems, Encyclopedia Math. Appl. 162, Cambridge University, Cambridge, 2016. · Zbl 1356.49003
[14] E. Shamir, Regularization of mixed second-order elliptic problems, Israel J. Math. 6 (1968), 150-168. · Zbl 0157.18202
[15] G. Stampacchia, Problemi al contorno ellitici, con dati discontinui, dotati di soluzionie Hölderiane, Ann. Mat. Pura Appl. (4) 51 (1960), 1-37. · Zbl 0204.42001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.