Bounded $H^\infty$-calculus for a Degenerate Elliptic Boundary Value Problem

TL;DR

This paper establishes that certain degenerate elliptic boundary value problems on manifolds with boundary have a bounded $H^$-calculus, extending functional calculus results to more general degenerate operators.

Contribution

It proves the bounded $H^$-calculus property for the $L_p$-realization of a class of degenerate elliptic boundary value problems on manifolds with boundary.

Findings

Bounded $H^$-calculus established for degenerate elliptic operators.

Results apply to operators with boundary degeneracies of a specific form.

Extends functional calculus theory to manifolds with boundary and degenerate operators.

Abstract

On a manifold $X$ with boundary and bounded geometry we consider a strongly elliptic second order operator $A$ together with a degenerate boundary operator $T$ of the form $T=\varphi_0\gamma_0 + \varphi_1\gamma_1$. Here $\gamma_0$ and $\gamma_1$ denote the evaluation of a function and its exterior normal derivative, respectively, at the boundary. We assume that $\varphi_0,\varphi_1\in C^{\infty}_b(\partial X)$, $\varphi_0,\varphi_1\ge 0$, and $\varphi_0+\varphi_1\geq c$, for some $c>0$. We also assume that the highest order coefficients of $A$ belong to $C^\tau(X)$ for some $\tau>0$ and the lower order coefficients are in $L_\infty(X)$. We show that the $L_p(X)$-realization of $A$ which respect to the boundary operator $T$ has a bounded $H^\infty$-calculus.