Existence of Variational Solutions in Noncylindrical Domains

Research output: Contribution to journalArticleScientificpeer-review

Researchers

  • Verena Boegelein
  • Frank Duzaar
  • Christoph Scheven
  • Thomas Singer

Research units

  • University of Salzburg
  • University of Duisburg-Essen
  • Friedrich-Alexander University Erlangen-Nürnberg

Abstract

We study gradient flows of integral functionals in noncylindrical bounded domains E subset of R-n [0, T). The systems of differential equations take the form partial derivative(t)u - divD(xi)f (x, u, Du) = -D(u)f (x, u, Du) on E, for an integrand f(x, u, Du) that is convex and coercive with respect to the W-1,W-P-norm for p > 1. We prove the existence of variational solutions on noncylindrical domains under the only assumption that Ln+1(partial derivative E) = 0, even for functionals that do not admit a growth condition from above. For nondecreasing domains, the solutions are unique and admit a time-derivative in L-2(E). For domains that decrease the most with bounded speed and integrands that satisfy a p-growth condition, we prove that the constructed solutions are continuous in time with respect to the L-2-norm and solve the above system of differential equations in the weak sense. Under the additional assumption that the domain also increases the most at finite speed, we establish the uniqueness of solutions.

Details

Original languageEnglish
Pages (from-to)3007-3057
Number of pages51
JournalSIAM Journal on Mathematical Analysis
Volume50
Issue number3
Publication statusPublished - 2018
MoE publication typeA1 Journal article-refereed

    Research areas

  • parabolic systems, variational solutions, noncylindrical domains, existence, continuity, LINEAR PARABOLIC EQUATION, PATTERN-FORMATION, DEGENERATE, INEQUALITIES, REGULARITY, CRITERION, CALCULUS, SOBOLEV, FLOW

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