Existence of variational solutions in noncylindrical domains

Bögelein V, Duzaar F, Scheven C, Singer T (2018)

Publication Type: Journal article

Publication year: 2018


Book Volume: 50

Pages Range: 3007--3057

Journal Issue: 3

DOI: 10.1137/17M1156423


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.

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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


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

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