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Register a new userMultiscale homogenization of integral convex functionals in Orlicz Sobolev setting
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The $Gamma $-limit of a family of functionals $umapsto int_{Omega }fleft( frac{x}{varepsilon },frac{x}{varepsilon ^{2}},D^{s}uright) dx$ is obtained for $s=1,2$ and when the integrand $f=fleft( y,z,vright) $ is a continous function, periodic in $y$ and $z$ and convex with respect to $v$ with nonstandard growth. The reiterated two-scale limits of second order derivative are characterized in this setting.
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We consider shape optimization problems for general integral functionals of the calculus of variations, defined on a domain $Omega$ that varies over all subdomains of a given bounded domain $D$ of ${bf R}^d$. We show in a rather elementary way the existence of a solution that is in general a quasi open set. Under very mild conditions we show that the optimal domain is actually open and with finite perimeter. Some counterexamples show that in general this does not occur.
We consider shape optimization problems for general integral functionals of the calculus of variations that may contain a boundary term. In particular, this class includes optimization problems governed by elliptic equations with a Robin condition on the free boundary. We show the existence of an optimal domain under rather general assumptions and we study the cases when the optimal domains are open sets and have a finite perimeter.
We provide relaxation for not lower semicontinuous supremal functionals of the type $W^{1,infty}(Omega;mathbb R^d) i u mapstosupess_{ x in Omega}f( abla u(x))$ in the vectorial case, where $Omegasubset mathbb R^N$ is a Lipschitz, bounded open set, and $f$ is level convex. The connection with indicator functionals is also enlightened, thus extending previous lower semicontinuity results in that framework. Finally we discuss the $L^p$-approximation of supremal functionals, with non-negative, coercive densities $f=f(x,xi)$, which are only $L^N otimes B_{d times N}$-measurable.
Let $H(q,p)$ be a Hamiltonian on $T^*T^n$. We show that the sequence $H_{k}(q,p)=H(kq,p)$ converges for the $gamma$ topology defined by the author, to $bar{H}(p)$. This is extended to the case where only some of the variables are homogenized, that is the sequence $H(kx,y,q,p)$ where the limit is of the type ${bar H}(y,q,p)$ and thus yields an effective Hamiltonian. We give here the proof of the convergence, and the first properties of the homogenization operator, and give some immediate consequences for solutions of Hamilton-Jacobi equations, construction of quasi-states, etc. We also prove that the function $bar H$ coincides with Mathers $alpha$ function which gives a new proof of its symplectic invariance proved by P. Bernard. A previous version of this paper relied on the former On the capacity of Lagrangians in $T^*T^n$ which has been withdrawn. The present version of Symplectic Homogenization does not rely on it anymore.
In this paper we build up a criteria for fractional Orlicz-Sobolev extension and imbedding domains on Ahlfors $n$-regular domains.
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