TY - JOUR

T1 - Aleksandrov reflection and nonlinear evolution equations, I

T2 - The n-sphere and n-ball

AU - Chow, Bennett

AU - Gulliver, Robert

N1 - Copyright:
Copyright 2017 Elsevier B.V., All rights reserved.

PY - 1996/4

Y1 - 1996/4

N2 - 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).

AB - 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).

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U2 - 10.1007/BF01254346

DO - 10.1007/BF01254346

M3 - Article

AN - SCOPUS:0039679681

SN - 0944-2669

VL - 4

SP - 249

EP - 264

JO - Calculus of Variations and Partial Differential Equations

JF - Calculus of Variations and Partial Differential Equations

IS - 3

ER -