Existence of variational solutions in noncylindrical domains

Beitrag in einer Fachzeitschrift


Details zur Publikation

Autorinnen und Autoren: Bögelein V, Duzaar F, Scheven C, Singer T
Zeitschrift: SIAM Journal on Mathematical Analysis
Jahr der Veröffentlichung: 2018
Band: 50
Heftnummer: 3
Seitenbereich: 3007--3057
ISSN: 0036-1410


Abstract

We study gradient flows of integral functionals in noncylindrical bounded domains E ⊂ Rⁿ × [0, T). The systems of differential equations take the form ∂tu − divDξf(x, u, Du) = −Duf(x, u, Du) on E, for an integrand f(x, u, Du) that is convex and coercive with respect to the W1, p-norm for p > 1. We prove the existence of variational solutions on noncylindrical domains under the only assumption that Lⁿ⁺¹(∂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 L²(E). For domains that decrease the most with bounded speed and integrands that satisfy a pgrowth condition, we prove that the constructed solutions are continuous in time with respect to the L²-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.


FAU-Autorinnen und Autoren / FAU-Herausgeberinnen und Herausgeber

Duzaar, Frank Prof. Dr.
Lehrstuhl für Mathematik (Partielle Differentialgleichungen und Variationsrechnung)
Singer, Thomas
Department Mathematik


Einrichtungen weiterer Autorinnen und Autoren

Universität Duisburg-Essen (UDE)
Universität Salzburg (Paris Lodron Universität Salzburg)


Zitierweisen

APA:
Bögelein, V., Duzaar, F., Scheven, C., & Singer, T. (2018). Existence of variational solutions in noncylindrical domains. SIAM Journal on Mathematical Analysis, 50(3), 3007--3057. https://dx.doi.org/10.1137/17M1156423

MLA:
Bögelein, Verena, et al. "Existence of variational solutions in noncylindrical domains." SIAM Journal on Mathematical Analysis 50.3 (2018): 3007--3057.

BibTeX: 

Zuletzt aktualisiert 2019-15-01 um 13:50