Interior estimates of harmonic heat flow

Motivated by Giaquinta and Hildebrandt’s regularity result for harmonic mappings [M. Giaquinta and S. Hildebrandt, A priori estimates for harmonic mappings, J. Reine Angew. Math. 1982(336) (1982) 124–164, Theorems 3 and 4], we show a [Formula: see text]-regularity result of the harmonic flow between two Riemannian manifolds when the image is in a regular geodesic ball. The proof is based on De Giorgi–Moser’s iteration and Schauder estimate.

Existence of Solutions

This chapter proves existence of solutions to the inhomogeneous problem using the Schauder estimate and analyzes a generalized Kimura diffusion operator, L, defined on a manifold with corners, P. The discussion centers on the solution w = v + u, where v solves the homogeneous Cauchy problem with v(x, 0) = f(x) and u solves the inhomogeneous problem with u(x, 0) = 0. The chapter first provides definitions for the Wright–Fisher–Hölder spaces on a general compact manifold with corners before explaining the steps involved in the existence proof. It then verifies the induction hypothesis and treats the k = 0 case. It also shows how to perform the doubling construction for P and considers the existence of the resolvent operator and a contraction semi-group. Finally, it discusses the problem of higher regularity.

Linear second order elliptic equations with Venttsel boundary conditions

SynopsisWe prove a Schauder estimate for solutions of linear second order elliptic equations with linear Venttsel boundary conditions, and establish an existence result for classical solutions for such boundary value problems.