### ${\mathcal{L}}^{2,\text{\Phi}}$ regularity for nonlinear elliptic systems of second order.

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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.

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....

Starting from Giaquinta’s counterexample [12] we introduce the class of splitting functionals being of $(p,q)$-growth with exponents $p\le q\<\infty $ and show for the scalar case that locally bounded local minimizers are of class ${C}^{1,\mu}$. Note that to our knowledge the only ${C}^{1,\mu}$-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.

Let $L:{\mathbb{R}}^{N}\times {\mathbb{R}}^{N}\to \mathbb{R}$ be a borelian function and consider the following problems$$inf\left\{F\left(y\right)={\int}_{a}^{b}L(y\left(t\right),{y}^{\text{'}}\left(t\right))\phantom{\rule{0.166667em}{0ex}}\mathrm{d}t:\phantom{\rule{0.166667em}{0ex}}y\in AC([a,b],{\mathbb{R}}^{N}),y\left(a\right)=A,\phantom{\rule{0.166667em}{0ex}}y\left(b\right)=B\right\}\phantom{\rule{2.0em}{0ex}}\phantom{\rule{1.0em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\left(P\right)$$$$\phantom{\rule{-17.07182pt}{0ex}}inf\left\{{F}^{...}\right\}$$

Let $L:{\mathbb{R}}^{N}\times {\mathbb{R}}^{N}\to \mathbb{R}$ be a Borelian function and consider the following problems $$inf\left\{F\left(y\right)={\int}_{a}^{b}L(y\left(t\right),{y}^{\text{'}}\left(t\right))\phantom{\rule{0.166667em}{0ex}}\mathrm{d}t:\phantom{\rule{0.166667em}{0ex}}y\in AC([a,b],{\mathbb{R}}^{N}),y\left(a\right)=A,\phantom{\rule{0.166667em}{0ex}}y\left(b\right)=B\right\}\phantom{\rule{2.0em}{0ex}}\phantom{\rule{1.0em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\left(P\right)$$ $$inf\left\{{F}^{**}\left(y\right)={\int}_{a}^{b}{L}^{**}(y\left(t\right),{y}^{\text{'}}\left(t\right))\phantom{\rule{0.166667em}{0ex}}\mathrm{d}t:\phantom{\rule{0.166667em}{0ex}}y\in AC([a,b],{\mathbb{R}}^{N}),y\left(a\right)=A,\phantom{\rule{0.166667em}{0ex}}y\left(b\right)=B\right\}\xb7\phantom{\rule{1.0em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\left({P}^{**}\right)$$ We give a sufficient condition, weaker then superlinearity, under which $infF=inf{F}^{**}$ 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.

We prove higher integrability for the gradient of bounded minimizers of some variational integrals with anisotropic growth.

In this note we study the summability properties of the minima of some non differentiable functionals of Calculus of the Variations.

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 $\partial Z$ of $Z$. We prove that in the regular case, this condition is also necessary.

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 ${I}_{}\left(u\right)=\frac{1}{2}{\int}_{\Omega}^{-1}{|1-|Du|}^{2}{|}^{2}+{\left|{D}^{2}u\right|}^{2}\mathrm{d}z$ 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 ${W}_{0}^{2,2}\left(\Omega \right)$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....

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 ${\mathit{I}}_{\mathit{\u03f5}}\mathrm{\left(}\mathit{u}\mathrm{\right)}\mathrm{=}\frac{\mathrm{1}}{\mathrm{2}}{{}^{\mathrm{\int}}}_{\mathit{\Omega}}{\mathit{\u03f5}}^{-1}{\left|\mathrm{1}\mathrm{-}{\left|\mathit{Du}\right|}^{\mathrm{2}}\right|}^{\mathrm{2}}\mathrm{+}\mathit{\u03f5}{\left|{\mathit{D}}^{\mathrm{2}}\mathit{u}\right|}^{\mathrm{2}}\mathrm{d}\mathit{z}$ where u belongs to the subset of functions in ${\mathit{W}}_{\mathrm{0}}^{\mathrm{2}\mathit{,}\mathrm{2}}\mathrm{\left(}\mathit{\Omega}\mathrm{\right)}$ whose gradient (in the sense of trace) satisfies Du(x)·ηx = 1 where ηx is the inward pointing unit normal ...

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.