Existence of Variational Solutions in Noncylindrical Domains

Loading...
Thumbnail Image
Access rights
openAccess
Journal Title
Journal ISSN
Volume Title
A1 Alkuperäisartikkeli tieteellisessä aikakauslehdessä
This publication is imported from Aalto University research portal.
View publication in the Research portal
View/Open full text file from the Research portal
Date
2018
Major/Subject
Mcode
Degree programme
Language
en
Pages
51
3007-3057
Series
SIAM JOURNAL ON MATHEMATICAL ANALYSIS, Volume 50, issue 3
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.
Description
Keywords
parabolic systems, variational solutions, noncylindrical domains, existence, continuity, LINEAR PARABOLIC EQUATION, PATTERN-FORMATION, DEGENERATE, INEQUALITIES, REGULARITY, CRITERION, CALCULUS, SOBOLEV, FLOW
Other note
Citation
Boegelein , V , Duzaar , F , Scheven , C & Singer , T 2018 , ' Existence of Variational Solutions in Noncylindrical Domains ' , SIAM Journal on Mathematical Analysis , vol. 50 , no. 3 , pp. 3007-3057 . https://doi.org/10.1137/17M1156423