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- ItemPohozaev identities for anisotropic integro-differential operators(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Ros-Oton, Xavier; Serra, Joaquim; Valdinoci, EnricoWe 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.
- ItemImprovement of flatness for nonlocal phase transitions(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Dipierro, Serena; Serra, Joaquim; Valdinoci, EnricoWe prove an improvement of flatness result for nonlocal phase transitions. For a class of nonlocal equations, we obtain a result in the same spirit of a celebrated theorem of Savin for the classical case. The results presented are completely new even for the case of the fractional Laplacian, but the robustness of the proofs allows us to treat also more general, possibly anisotropic, integro-differential operators.
- ItemNonlocal phase transitions: Rigidity results and anisotropic geometry(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Dipierro, Serena; Serra, Joaquim; Valdinoci, EnricoFinally, we consider a nonlocal equation with a multiwell potential, motivated by models arising in crystal dislocations, and we construct orbits exhibiting symbolic dynamics, inspired by some classical results by Paul Rabinowitz.