Operator-valued Symbols for Elliptic and Parabolic Problems on Wedges

We study evolution problems of the type e∂tu+h(∂x)u = f where h is a holomorphic function on a vertical strip around the imaginary axis, and s > 0. If P is a second-order polynomial we give a complete characterization of the spectrum of the parameter-dependent operator λe +P (∂x) in Lp(R). We show the surprising result that the spectrum is independent of λ whenever | arg λ| < π. Moreover, we also characterize the spectrum of ∂te + P (∂x), and we show that this operator admits a bounded H∞-calculus. Finally, we describe applications to free boundary problems with moving contact lines, and we study the diffusion equation in an angle or a wedge domain with dynamic boundary conditions. Our approach relies on the H∞-calculus for sectorial operators, the concept of R-boundedness, and recent results for the sum of non-commuting operators.