Bounded $$H^\infty $$-calculus for a degenerate elliptic boundary value problem

Abstract On a manifold X with boundary and bounded geometry we consider strongly elliptic second order operator A together degenerate T of the form $$T=\varphi _0\gamma _0 + \varphi _1\gamma _1$$ T=φ0γ0+φ1γ1 . Here $$\gamma _0$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">γ0 xmlns:mml="http://www.w3.org/1998/Math/MathML">γ1 denote evaluation function its exterior normal derivative, respectively, at boundary. We assume that $$\varphi _0, _1\ge 0$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">φ0,φ1≥0 , _0+\varphi c$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">φ0+φ1≥c for some $$c>0$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">c>0 where either _0,\varphi _1\in C^{\infty }_b(\partial X)$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">φ0,φ1∈Cb∞(∂X) or _0=1 $$ xmlns:mml="http://www.w3.org/1998/Math/MathML">φ0=1 _1=\varphi ^2$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">φ1=φ2 \in C^{2+\tau }(\partial xmlns:mml="http://www.w3.org/1998/Math/MathML">φ∈C2+τ(∂X) $$\tau >0$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">τ>0 also highest coefficients belong to $$C^\tau (X)$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">Cτ(X) lower are in $$L_\infty xmlns:mml="http://www.w3.org/1998/Math/MathML">L∞(X) show $$L_p(X)$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">Lp(X) -realization respect has $$H^\infty xmlns:mml="http://www.w3.org/1998/Math/MathML">H∞ -calculus. then obtain unique solvability associated value problem adapted spaces. As an application, short time existence solutions porous medium equation.