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Archive for Rational Mechanics and Analysis

, Volume 210, Issue 3, pp 975–1020 | Cite as

Nonlinear Elliptic–Parabolic Problems

  • Inwon C. Kim
  • Norbert Požár
Article

Abstract

We introduce a notion of viscosity solutions for a general class of elliptic–parabolic phase transition problems. These include the Richards equation, which is a classical model in filtration theory. Existence and uniqueness results are proved via the comparison principle. In particular, we show existence and stability properties of maximal and minimal viscosity solutions for a general class of initial data. These results are new, even in the linear case, where we also show that viscosity solutions coincide with the regular weak solutions introduced in Alt and Luckhaus (Math Z 183:311–341, 1983).

Keywords

Weak Solution Viscosity Solution Elliptic Problem Comparison Principle Parabolic Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Department of MathematicsUCLALos AngelesUSA
  2. 2.Graduate School of Mathematical SciencesUniversity of TokyoTokyoJapan

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