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.
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.
B. Barrios and M. Medina, Strong maximum principles for fractional elliptic and parabolic problems with mixed boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A, https://doi.org/10.1017/prm.2018.77.
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.
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.
L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations 32 (2007), no. 7–9, 1245–1260.
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.
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.
E. Colorado and I. Peral, Semilinear elliptic problems with mixed Dirichlet–Neumann boundary conditions, J. Funct. Anal. 199 (2003), no. 2, 468–507.
S. Dipierro, X. Ros-Oton and E. Valdinoci, Nonlocal problems with Neumann boundary conditions, Rev. Mat. Iberoam. 33 (2017), no. 2, 377–416.
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.
D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Pure Appl. Math. 88, Academic Press, New York, 1980.
J.-L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications. Vol. I, Grundlehren Math. Wiss. 181, Springer, New York, 1972,
G. Molica Bisci, V. D. Radulescu and R. Servadei, Variational Methods for Nonlocal Fractional Problems, Encyclopedia Math. Appl. 162, Cambridge University, Cambridge, 2016.
E. Shamir, Regularization of mixed second-order elliptic problems, Israel J. Math. 6 (1968), 150–168.
G. Stampacchia, Problemi al contorno ellitici, con dati discontinui, dotati di soluzionie Hölderiane, Ann. Mat. Pura Appl. (4) 51 (1960), 1–37.
E. Colorado and A. Ortega are partially supported by the Ministry of Economy and Competitiveness of Spain and FEDER, under grant number MTM2016-80618-P. J. Carmona is partially supported by Ministerio de Ciencia, Innovación y Universidades (MCIU), Agencia Estatal de Investigación (AEI) and Fondo Europeo de Desarrollo Regional (FEDER) under grant number PGC2018-096422-B-I00 and Junta de Andalucía FQM-194.
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