Displaying similar documents to “Higher order variational problems on two-dimensional domains.”

On the regularity of local minimizers of decomposable variational integrals on domains in ℝ 2

Michael Bildhauer, Martin Fuchs (2007)

Commentationes Mathematicae Universitatis Carolinae

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We consider local minimizers u : ℝ 2 ⊃ Ω → ℝ N of variational integrals like ∫ Ω [ ( 1 + | ∂ 1 u | 2 ) p / 2 + ( 1 + | ∂ 2 u | 2 ) q / 2 ] d x or its degenerate variant ∫ Ω [ | ∂ 1 u | p + | ∂ 2 u | q ] d x with exponents 2 ≤ p < q < ∞ which do not fall completely in the category studied in Bildhauer M., Fuchs M., Calc. Var. (2003), 177–186. We prove interior C 1 , α - respectively C 1 -regularity of u under the condition that q < 2 p . For decomposable variational integrals of arbitrary order a similar result is established by the way extending the work Bildhauer M., Fuchs M., Ann. Acad. Sci. Fenn. Math. (2006), 349–362. ...

Smoothness properties of solutions to the nonlinear Stokes problem with nonautonomous potentials

Dominic Breit (2013)

Commentationes Mathematicae Universitatis Carolinae

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We discuss regularity results concerning local minimizers u : ℝ n ⊃ Ω → ℝ n of variational integrals like ∫ Ω { F ( · , ε ( w ) ) - f · w } d x defined on energy classes of solenoidal fields. For the potential F we assume a ( p , q ) -elliptic growth condition. In the situation without x -dependence it is known that minimizers are of class C 1 , α on an open subset Ω 0 of Ω with full measure if q < p n + 2 n (for n = 2 we have Ω 0 = Ω ). In this article we extend this to the case of nonautonomous integrands. Of course our result extends to weak solutions of the corresponding...