Smooth Contractive Embeddings and Application to Feynman Formula for Parabolic Equations on Smooth Bounded Domains

TL;DR

This paper verifies key assumptions for the existence of a semigroup solving a boundary value problem for elliptic operators, enabling a Feynman formula for solutions on smooth bounded domains.

Contribution

It proves the assumptions hold for domains with $C^{4,eta}$ boundary and $C^{2,eta}$ coefficients, advancing the theoretical foundation for Feynman formulas.

Findings

Assumptions are valid for $C^{4,eta}$-smooth domains.

Supports construction of Feynman formulas for elliptic PDEs.

Provides a rigorous basis for semigroup solutions in smooth domains.

Abstract

We prove two assumptions made in an article by Ya.A. Butko, M. Grothaus, O.G. Smolyanov concerning the existence of a strongly continuous operator semigroup solving a Cauchy-Dirichlet problem for an elliptic differential operator in a bounded domain and the existence of a smooth contractive embedding of a core of the generator of the semigroup into the space $C_c^{2,\alpha}(\R^n)$. Based on these assumptions a Feynman formula for the solution of the Cauchy-Dirichlet problem is constructed in the article mentioned above. In this article we show that the assumptions are fulfilled for domains with $C^{4,\alpha}$-smooth boundary and coefficients in $C^{2,\alpha}$.