Semigroups, Boundary Value Problems and Markov Processes

Green and Poisson functions with Wentzell boundary conditions

“…For instance, one finds existence and uniqueness results for Dirichlet (and sometimes Neumann) boundary conditions in Bensoussan and Lions [3], while the construction and estimates for the Green function with oblique boundary conditions can be found in [6,7]. Also recently, e.g., see Favini et al [23], Taira [22].…”

Semilinear degenerate elliptic boundary value problems at resonance

“…then it follows from an application of [15,Theorem 9.1] with ϕ := 0 that the operator A is a Fredholm operator with index zero:…”

“…More precisely, condition (H.1) implies that the absorption phenomenon occurs at each point of the set M = {x ∈ ∂ : a(x ) = 0}, while the reflection phenomenon occurs at each point of the set ∂ \M = {x ∈ ∂ : a(x ) > 0}. In other words, a Markovian particle moves continuously in the space \M until it dies at the time when it reaches the set M where the particle is definitely absorbed (see [15]). On the other hand, condition (H.2) implies that the boundary condition B is not equal to the purely Neumann condition.…”

Elliptic problems with nonlocal boundary conditions and Feller semigroups

“…In the nontransversal case, these orders coincide. Sato and Ueno [76], Bony, Courrege, and Priouret [10], Watanabe [109], Taira [101][102][103], Ishikawa [48], and others studied the transversal case. Skubachevskii [86,91,92] proposed a method of studying the nontransversal case.…”