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A parabolic analogue of Finn’s maximum principle. (English) Zbl 0977.35073

The author extends to the parabolic equation \[ u_t=\sum^n_{i=1} \frac{\partial}{\partial x_i} \frac{u_{x_i}}{\sqrt{1+|\nabla_x u|^2}}\tag{1} \] a comparison principle for the minimal surface equation established by the reviewer [J. Anal. Math. 14, 139-160 (1965; Zbl 0163.34604)]. Specifically, let \(D\in\mathbb{R}^n\), \(D_T = (0,T)\times D\), let \(E\) denote the exterior of the sphere \(\{r=a\}\) in \(\mathbb{R}^n\), \(E_T=(0,T)\times E\), \(\partial_FD_T=D\cup \{[0,T)\times \{\partial D\setminus\{r=a\}\}\}\). Let \(c_1\geq 1\) and \(c\) be arbitrary constants. The author obtains:
Theorem: Let \(u(t,x)\) be a (generalized) subsolution of (1) in \(D_T\subseteq E_T\), continuous on \(D_T\cup \partial_FD_T\), and such that \(u - c_1 \text{cosh}^{-1}(\frac ra)\leq c\) on \(\partial_FD_F\). Then \(u -c_1\text{cosh}^{-1}(\frac ra)\leq c\) in \(D_T\).
If \(u\) is a supersolution, then the same result holds with inequalities reversed and with the negative signs replaced by positive signs.
The author points out some striking consequences of the result, that are analogous to the known properties of solutions to the minimal surface equation, and which have no analogues in the theory of linear equations. The demonstration is obtained with the aid of a comparison principle that again depends essentially on the particular nonlinearity.

MSC:

35K65 Degenerate parabolic equations
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs

Citations:

Zbl 0163.34604
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