We study the interior regularity of solutions to the Dirichlet problem Lu = g in Omega, u = 0 in R-nOmega, for anisotropic operators of fractional type Lu(x) = integral(+infinity)(0) dp integral(Sn-1) da(w) 2u(x) - u(x + rho w) - u(x - rho w)/rho(1+2s). Here, a is any measure on Sn-1 (a prototype example for L is given by the sum of one-dimensional fractional Laplacians in fixed, given directions). When a is an element of C-infinity(Sn-1) and g is c(infinity)(Omega), solutions are known to be C-infinity inside Omega (but not up to the boundary). However, when a is a general measure, or even when a is L-infinity(s(n-1)), solutions are only known to be C-3s inside Omega. We prove here that, for general measures a, solutions are C1+3s-epsilon inside Omega for all epsilon > 0 whenever Omega is convex. When a is an element of L-infinity(Sn-1), we show that the same holds in all C-1,C-1 domains. In particular, solutions always possess a classical first derivative. The assumptions on the domain are sharp, since if the domain is not convex and the measure a is singular, we construct an explicit counterexample for which u is not C3s+epsilon for any epsilon > 0 - even if g and Omega are C-infinity.
The Dirichlet problem for nonlocal operators with singular kernels: convex and nonconvex domains / X. Ros Oton, E. Valdinoci. - In: ADVANCES IN MATHEMATICS. - ISSN 0001-8708. - 288(2016), pp. 732-790.
|Titolo:||The Dirichlet problem for nonlocal operators with singular kernels: convex and nonconvex domains|
|Parole Chiave:||Regularity theory; Integro-differential equations; Fractional Laplacian; Anisotropic media; Rough kernels|
|Settore Scientifico Disciplinare:||Settore MAT/05 - Analisi Matematica|
|Data di pubblicazione:||2016|
|Data ahead of print / Data di stampa:||18-nov-2015|
|Digital Object Identifier (DOI):||http://dx.doi.org/10.1016/j.aim.2015.11.001|
|Appare nelle tipologie:||01 - Articolo su periodico|