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

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 $T={\phi }_0{\gamma }_0+{\phi }_1{\gamma }_1$. Here ${\gamma }_0$ and ${\gamma }_1$ denote the evaluation of a function and its exterior normal derivative, respectively, at the boundary. We assume that ${\phi }_0,{\phi }_1\in C_b^{\infty }(\partial X)$, ${\phi }_0,{\phi }_1\ge 0$, and ${\phi }_0+{\phi }_1\ge c$, 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.