Core property of smooth contractive embeddable functions for an elliptic operator

TL;DR

This paper studies the properties of elliptic differential operators with smooth coefficients, showing they generate semigroups on boundary conditions and constructing smooth, contractive extension operators for functions satisfying these conditions.

Contribution

It establishes the core property of smooth contractive embeddable functions for elliptic operators and constructs a linear extension operator preserving smoothness and boundary conditions.

Findings

Elliptic operators with smooth coefficients generate strongly continuous semigroups.

The subspace of smooth functions satisfying boundary conditions forms a core for the semigroup generator.

A linear extension operator preserves smoothness and boundary conditions without increasing the supremum.

Abstract

Given an elliptic differential operator L of second order with smooth coefficients in a bounded domain with smooth boundary. We show that if the coefficients are H\"older-continuous up to the boundary and the boundary is $C^{2,\alpha}$-smooth that on the space of all $C^{2,\alpha}$-smooth (up to the boundary) functions u fulfilling both u=0 and Lu=0 (on the boundary) the operator L is dissipative and closable to an generator of a strong continuous operator semigroup in the space of continuous functions with zero boundary condition. Moreover we show that if the coefficients of the second order and first order derivatives are in $C^{2,\alpha}$ then the above mentioned subspace of $C^{2,\alpha}$ which is a core for the generator of the semigroup, can be embedded (continued) in a contractive and smooth way. Thus we construct a linear extension operator which maps a $C^{2,\alpha}$-smooth function fulfilling the boundary conditions to a $C^{2,\alpha}$-smooth function in $\R^n$ such that the supremum is not increased.