Calculus of Variations and Geometric Measure Theory
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D. Kwon - A. R. Mészáros

Degenerate nonlinear parabolic equations with discontinuous diffusion coefficients

created by mészáros on 03 Aug 2019

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Submitted Paper

Inserted: 3 aug 2019
Last Updated: 3 aug 2019

Year: 2019

Abstract:

This paper is devoted to the study of some nonlinear parabolic equations with discontinuous diffusion intensities. Such problems appear naturally in physical and biological models. Our analysis is based on variational techniques and in particular on gradient flows in the space of probability measures equipped with the distance arising in the Monge-Kantorovich optimal transport problem. The associated internal energy functionals in general fail to be differentiable and geodesically $\lambda$-convex, therefore classical results do not apply in our setting. We study the combination of both linear and porous medium type diffusions and we show the existence and uniqueness of the solutions in the sense of distributions in suitable Sobolev spaces. Our notion of solution allows us to give a fine characterization of the emerging critical regions, observed previously in numerical experiments. A link to a three phase free boundary problem is also pointed out.


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