Aleksandrov reflection and nonlinear evolution equations, I: The n-sphere and n-ball

Bennett Chow, Robert Gulliver

Research output: Contribution to journalArticlepeer-review

39 Scopus citations

Abstract

We consider the (degenerate) parabolic equation ut = G(∇∇u + ug, t) on the n-sphere Sn. This corresponds to the evolution of a hypersurface in Euclidean space by a general function of the principal curvatures, where u is the support function. Using a version of the Aleksandrov reflection method, we prove the uniform gradient estimate |∇u(·, t)| ≤ C, where C depends on the initial condition u(·, 0) but not on t, nor on the nonlinear function G. We also prove analogous results for the equation ut = G(Δu + cu, |x|, t) on the n-ball Bn, where c ≤ λ2(Bn).

Original languageEnglish (US)
Pages (from-to)249-264
Number of pages16
JournalCalculus of Variations and Partial Differential Equations
Volume4
Issue number3
DOIs
StatePublished - Apr 1996

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