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Pohozaev identities for anisotropic integro-differential operators
We establish Pohozaev identities and integration by parts type formulas for anisotropic integro-differential operators of order 2s, with s ϵ (0, 1). These identities involve local boundary terms, in which the quantity u/ds ∂Ω plays the role that ∂u/∂v plays in the second order case. Here, u is any solution to Lu = f (x, u) in Ω, with u = 0 in Rn \ Ω , and d is the distance to ∂Ω.
Nonlocal problems with Neumann boundary conditions
We introduce a new Neumann problem for the fractional Laplacian arising from a simple probabilistic consideration, and we discuss the basic properties of this model. We can consider both elliptic and parabolic equations in any domain. In addition,we formulate problems with nonhomogeneous Neumann conditions, and also with mixed Dirichlet and Neumann conditions, all of them having a clear probabilistic interpretation. We prove that solutions to the fractional heat equation with homogeneous Neumann conditions have the following natural properties: conservation of mass, decreasing energy, and convergence to a constant as time flows. Moreover, for the elliptic case we give the variational formulation of the problem, and establish existence of solutions. We also study the limit properties and the boundary behavior induced by this nonlocal Neumann condition.
The Dirichlet problem for nonlocal operators with kernels: Convex and nonconvex domains
We study the interior regularity of solutions to a Dirichlet problem for anisotropic operators of fractional type. A prototype example is given by the sum of one-dimensional fractional Laplacians in fixed, given directions. We prove here that an interior differentiable regularity theory holds in convex domains. When the spectral measure is a bounded function and the domain is smooth, the same regularity theory applies. 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 spectral measure is singular, we construct an explicit counterexample.