Solvability of linear local and nonlocal Robin problems over

Let Ω ⊆ R N be a bounded Lipschitz domain. We first consider an elliptic boundary value problem with general Robin boundary conditions. The boundary conditions can be either local or nonlocal, depending on the conditions imposed on the elliptic operator. We prove that this boundary value problem is uniquely solvable, and moreover we show that such weak solution is Hölder continuous on Ω ¯ . We also prove that a realization of the associated differential operator with generalized local or nonlocal Robin boundary conditions generates an analytic C 0 -semigroup of angle π / 2 over C ( Ω ¯ ) . We conclude by applying the elliptic regularity theory to solve the corresponding Cauchy problem over C ( Ω ¯ ) .