Stationary isothermic surfaces and some characterizations of the hyperplane in the N-dimensional Euclidean space

Details Stationary isothermic surfaces and some characterizations of the hyperplane in the N-dimensional Euclidean space Stationary isothermic surfaces and some characterizations of the hyperplane in the N-dimensional Euclidean space

2008-12-11

arxiv.org/abs/0812.2165v1

Mathematics

Analysis of PDEs

11 pages

Scientific paper

We consider an entire graph S of a continuous real function over (N-1)-dimensional Euclidean space with N larger than or equal to 3. Let D be a domain in N-dimensional Euclidean space with S as a boundary. Consider in D the heat flow with initial temperature 0 and boundary temperature 1. The problem we consider is to characterize S in such a way that there exists a stationary isothermic surface in D. We show that S must be a hyperplane under some general conditions on S. This is related to Liouville or Bernstein-type theorems for some elliptic Monge-Amp\`ere-type equation.

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