Heat Kernel Asymptotics of Zaremba Boundary Value Problem

TL;DR

This paper investigates the heat kernel asymptotics for the Zaremba boundary value problem, which involves mixed boundary conditions on a smooth manifold, providing explicit computations of the leading terms and coefficients.

Contribution

It offers a detailed construction of the global parametrix for the heat equation and explicitly computes the first non-trivial coefficients of the asymptotic expansion.

Findings

Explicit formulas for the leading parametrix are derived.

First non-trivial heat kernel coefficients are computed.

The analysis extends understanding of mixed boundary conditions on manifolds.

Abstract

The Zaremba boundary-value problem is a boundary value problem for Laplace-type second-order partial differential operators acting on smooth sections of a vector bundle over a smooth compact Riemannian manifold with smooth boundary but with non-smooth (singular) boundary conditions, which include Dirichlet conditions on one part of the boundary and Neumann ones on another part of the boundary. We study the heat kernel asymptotics of Zaremba boundary value problem. The construction of the global parametrix of the heat equation is described in detail and the leading parametrix is computed explicitly. Some of the first non-trivial coefficients of the heat kernel asymptotic expansion are computed explicitly.