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A finite dimensional linear programming approximation of Mather's variational problem

Luca Granieri (2010)

ESAIM: Control, Optimisation and Calculus of Variations

We provide an approximation of Mather variational problem by finite dimensional minimization problems in the framework of Γ-convergence. By a linear programming interpretation as done in [Evans and Gomes, ESAIM: COCV 8 (2002) 693–702] we state a duality theorem for the Mather problem, as well a finite dimensional approximation for the dual problem.

A Mean Value Theorem for non Differentiable Mappings in Banach Spaces

Deville, Robert (1995)

Serdica Mathematical Journal

We prove that if f is a real valued lower semicontinuous function on a Banach space X and if there exists a C^1, real valued Lipschitz continuous function on X with bounded support and which is not identically equal to zero, then f is Lipschitz continuous of constant K provided all lower subgradients of f are bounded by K. As an application, we give a regularity result of viscosity supersolutions (or subsolutions) of Hamilton-Jacobi equations in infinite dimensions which satisfy a coercive condition....

A regularity theory for scalar local minimizers of splitting-type variational integrals

Michael Bildhauer, Martin Fuchs, Xiao Zhong (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

Starting from Giaquinta’s counterexample [12] we introduce the class of splitting functionals being of ( p , q ) -growth with exponents p q < and show for the scalar case that locally bounded local minimizers are of class C 1 , μ . Note that to our knowledge the only C 1 , μ -results without imposing a relation between p and q concern the case of two independent variables as it is outlined in Marcellini’s paper [15], Theorem A, and later on in the work of Fusco and Sbordone [10], Theorem 4.2.

A relaxation result for autonomous integral functionals with discontinuous non-coercive integrand

Carlo Mariconda, Giulia Treu (2010)

ESAIM: Control, Optimisation and Calculus of Variations

Let L : N × N be a Borelian function and consider the following problems inf F ( y ) = a b L ( y ( t ) , y ' ( t ) ) d t : y A C ( [ a , b ] , N ) , y ( a ) = A , y ( b ) = B ( P ) inf F * * ( y ) = a b L * * ( y ( t ) , y ' ( t ) ) d t : y A C ( [ a , b ] , N ) , y ( a ) = A , y ( b ) = B · ( P * * ) We give a sufficient condition, weaker then superlinearity, under which inf F = inf F * * if L is just continuous in x. We then extend a result of Cellina on the Lipschitz regularity of the minima of (P) when L is not superlinear.

A remark on the local Lipschitz continuity of vector hysteresis operators

Pavel Krejčí (2001)

Applications of Mathematics

It is known that the vector stop operator with a convex closed characteristic Z of class C 1 is locally Lipschitz continuous in the space of absolutely continuous functions if the unit outward normal mapping n is Lipschitz continuous on the boundary Z of Z . We prove that in the regular case, this condition is also necessary.

A simple proof of the characterization of functions of low Aviles Giga energy on a ball via regularity

Andrew Lorent (2012)

ESAIM: Control, Optimisation and Calculus of Variations

The Aviles Giga functional is a well known second order functional that forms a model for blistering and in a certain regime liquid crystals, a related functional models thin magnetized films. Given Lipschitz domain Ω ⊂ ℝ2the functional is I ( u ) = 1 2 Ω - 1 | 1 - | D u | 2 | 2 + | D 2 u | 2 d z I ϵ ( u ) = 1 2 ∫ Ω ϵ -1 1 − Du 2 2 + ϵ D 2 u 2 d z whereubelongs to the subset of functions in W 0 2 , 2 ( Ω ) W02,2(Ω) whose gradient (in the sense of trace) satisfiesDu(x)·ηx = 1 where ηx is the inward pointing unit normal to ∂Ω at x. In [Ann. Sc. Norm. Super. Pisa Cl....

A simple proof of the characterization of functions of low Aviles Giga energy on a ball via regularity

Andrew Lorent (2012)

ESAIM: Control, Optimisation and Calculus of Variations

The Aviles Giga functional is a well known second order functional that forms a model for blistering and in a certain regime liquid crystals, a related functional models thin magnetized films. Given Lipschitz domain Ω ⊂ ℝ2 the functional is I ϵ ( u ) = 1 2 Ω ϵ -1 1 Du 2 2 + ϵ D 2 u 2 d z where u belongs to the subset of functions in W 0 2 , 2 ( Ω ) whose gradient (in the sense of trace) satisfies Du(x)·ηx = 1 where ηx is the inward pointing unit normal ...

A Wiener type criterion for weighted quasiminima

Silvana Marchi (1991)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

We prove a sufficient condition of continuity at the boundary for quasiminima of degenerate type. W. P. Ziemer stated a Wiener-type criterion for the quasiminima defined by Giaquinta and Giusti. In this paper we extend the result of Ziemer to the case of weighted quasiminima, the weight being in the A 2 class of Muckenhoupt.

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