HEAT KERNEL BOUNDS FOR ELLIPTIC PARTIAL DIFFERENTIAL OPERATORS IN DIVERGENCE FORM WITH ROBIN-TYPE BOUNDARY CONDITIONS

摘要

One of the principal topics of this paper concerns the realization of self-adjoint operators L~2, in L~2(?; d~nx)~m, m, n ∈ N, associated with divergence form elliptic partial differential expressions L with (nonlocal) Robin-type boundary conditions in bounded Lipschitz domains ?? R~n. In particular, we develop the theory in the vector-valued case and hence focus on matrix-valued differential expressions L which act as The (nonlocal) Robin-type boundary conditions are then of the form where Θ represents an appropriate operator acting on Sobolev spaces associated with the boundary ?? of ?, ν denotes the outward pointing normal unit vector on ??, and Assuming Θ ≥ 0 in the scalar case m = 1, we prove Gaussian heat kernel bounds for L_(Θ,?) by employing positivity preserving arguments for the associated semigroups and reducing the problem to the corresponding Gaussian heat kernel bounds for the case of Neumann boundary conditions on ??. We also discuss additional zero-order potential coefficients V and hence operators corresponding to the form sum L_(Θ,?) + V.