Operator-valued Symbols for Elliptic and Parabolic Problems on Wedges
We study evolution problems of the type e sx ∂ t u+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 sx + P(∂ x) in L p(ℝ). We show the surprising result that the spectrum is independent of λ whenever |arg λ| < π. Moreover, we also characterize the spectrum of ∂ t e sx + 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.
KeywordsParabolic Problem Free Boundary Problem Sectorial Operator Maximal Regularity Kernel Operator
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