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Symmetry of solutions to parabolic Monge-Ampère equations

Limei Dai

Author Affiliations

School of Mathematics and Information Science, Weifang University, Weifang, Shandong, 261061, China

Boundary Value Problems 2013, 2013:185  doi:10.1186/1687-2770-2013-185

Published: 20 August 2013


In this paper, we study the parabolic Monge-Ampère equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M1">View MathML</a>

Using the method of moving planes, we show that any parabolically convex solution is symmetric with respect to some hyperplane. We also give a counterexample in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/185/mathml/M2">View MathML</a> and an example in a cylinder to illustrate the results.

MSC: 35K96, 35B06.

parabolic Monge-Ampère equations; symmetry; method of moving planes