Functional Inequalities for Doubly Weighted Brownian Motion with Sticky-reflecting Boundary Diffusion

Abstract We give upper bounds for the Poincaré and logarithmic Sobolev constants for doubly weighted Brownian motion on manifolds with sticky-reflecting boundary diffusion under curvature assumptions on the manifold and its boundary. To achieve this we use an interpolation approach based on energy interactions between the boundary and the interior of the manifold as well as the weighted Reilly formula. Along the way we also obtain a lower bound on the first nontrivial doubly weighted Steklov eigenvalue and an upper bound on the norm of the doubly weighted boundary trace operator on Sobolev functions. We also consider the case of doubly weighted Brownian motion with pure sticky reflection.