ABSTRACT
This chapter proves the last type of local Schauder estimates, up to a portion of the boundary, for solutions to parabolic equations on ∂ℝd+, which satisfy zero- or first-order boundary conditions. It also proves some other regularity results for the solution to the Cauchy problem. The chapter solves the Neumann Cauchy problem for the Laplacian, which is easier to analyze, since one can determine an explicit formula for its solution, and next using the method of continuity together with the a priori estimate. Using tools from the semigroup theory, it also solves an optimal spatial regularity result for solutions to problem.
