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Zbl 0923.35031
Kurta, V.V.
On analogues of Finn's maximum principle for solutions of parabolic equations. (English. Russian original)
[J] Russ. Math. Surv. 52, No.6, 1307-1309 (1997); translation from Usp. Mat. Nauk 52, No.6, 169-170 (1997). ISSN 0036-0279

One of the striking consequences of the nonlinearity in the minimal surface operator $${\cal M}u\equiv \sum^n_{i=1} {\partial\over\partial x_i} \Biggl({u_{x_i}\over \sqrt{1+|\nabla_x u|^2}}\Biggr)$$ is that if $u(x)$ defines a minimal surface over an annular domain ${\cal D}$, whose boundary contains a portion of the sphere $| x|= a$, and if $u(x)$ is bounded on the remaining part of the boundary of ${\cal D}$, then $u$ is bounded on all of ${\cal D}$. In this note, the author indicates that a similar method of proof yields an analogous property for parabolic operators $\partial_tu-{\cal M}u$.
[R.Finn (Stanford)]

MSC 2000:: *35B50 Maximum principles (PDE)
35K55 Nonlinear parabolic equations

Keywords: minimal surface operator; annular domain

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