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

Abstract

AbstractOn 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 $$T=\varphi _0\gamma _0 + \varphi _1\gamma _1$$ T = φ 0 γ 0 + φ 1 γ 1 . Here $$\gamma _0$$ γ 0 and $$\gamma _1$$ γ 1 denote the evaluation of a function and its exterior normal derivative, respectively, at the boundary. We assume that $$\varphi _0, \varphi _1\ge 0$$ φ 0 , φ 1 ≥ 0 , and $$\varphi _0+\varphi _1\ge c$$ φ 0 + φ 1 ≥ c , for some $$c>0$$ c > 0 , where either $$\varphi _0,\varphi _1\in C^{\infty }_b(\partial X)$$ φ 0 , φ 1 ∈ C b ∞ ( ∂ X ) or $$\varphi _0=1 $$ φ 0 = 1 and $$\varphi _1=\varphi ^2$$ φ 1 = φ 2 for some $$\varphi \in C^{2+\tau }(\partial X)$$ φ ∈ C 2 + τ ( ∂ X ) , $$\tau >0$$ τ > 0 . We also assume that the highest order coefficients of A belong to $$C^\tau (X)$$ C τ ( X ) and the lower order coefficients are in $$L_\infty (X)$$ L ∞ ( X ) . We show that the $$L_p(X)$$ L p ( X ) -realization of A with respect to the boundary operator T has a bounded $$H^\infty $$ H ∞ -calculus. We then obtain the unique solvability of the associated boundary value problem in adapted spaces. As an application, we show the short time existence of solutions to the porous medium equation.