Differential Operators on Conic Manifolds: Maximal Regularity and Parabolic Equations

Details Differential Operators on Conic Manifolds: Maximal Regularity and Parabolic Equations Differential Operators on Conic Manifolds: Maximal Regularity and Parabolic Equations

2002-01-20

arxiv.org/abs/math/0201184v2

Bull. Soc. Roy. Sci. Li\`ege 70, fasc. 4-5-6, 207-229 (2001)

Mathematics

Analysis of PDEs

18 pages (revised version, 23/04/'02)

Scientific paper

We study an elliptic differential operator A on a manifold with conic points. Assuming A to be defined on the smooth functions supported away from the singularities, we first address the question of possible closed extensions of A to L^p Sobolev spaces and then explain how additional ellipticity conditions ensure maximal regularity for the operator A. Investigating the Lipschitz continuity of the maps f(u)=|u|^\alpha, with real \alpha \ge 1, and f(u)=u^\alpha, with \alpha a natural number, and using a result of Cl\'ement and Li, we finally show unique solvability of a quasilinear equation of the form \dot{u} - a(u) \Delta u = f(u) in suitable spaces.

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