$\Gamma$-convergence techniques and relaxation results of constrained energy functionals are used to identify the limiting energy as the thickness $\epsilon$ approaches zero of a ferromagnetic thin structure ${\Omega }_{\epsilon }=\omega \times (-\epsilon ,\epsilon )$, $\omega \subset {ℝ}^2$, whose energy is given by$${ℰ}_{\epsilon }\left(\overline{m}\right)=\frac{1}{\epsilon }{\int }_{{\Omega }_{\epsilon }}\left(W(\overline{m},\nabla \overline{m})+\frac{1}{2}\nabla \overline{u}·\overline{m}\right)\mathrm{d}x$$subject to$$\text{div}(-\nabla \overline{u}+\overline{m}{\chi }_{{\Omega }_{\epsilon }})=0\text{on}{ℝ}^3,$$and to the constraint$$|\overline{m}|=1\text{on}{\Omega }_{\epsilon },$$where $W$ is any continuous function satisfying $p$-growth assumptions with $p\>1$. Partial results are also obtained in the case $p=1$, under an additional assumption on $W$.

Γ-convergence techniques and relaxation results of constrained energy functionals are used to identify the limiting energy as the thickness ε approaches zero of a ferromagnetic thin structure ${\Omega }_{\epsilon }=\omega \times (-\epsilon ,\epsilon )$, $\omega \subset {ℝ}^2$, whose energy is given by $${ℰ}_{\epsilon }\left(\overline{m}\right)=\frac{1}{\epsilon }{\int }_{{\Omega }_{\epsilon }}\left(W(\overline{m},\nabla \overline{m})+\frac{1}{2}\nabla \overline{u}·\overline{m}\right)\mathrm{d}x$$ subject to $$\text{div}(-\nabla \overline{u}+\overline{m}{\chi }_{{\Omega }_{\epsilon }})=0\text{on}{ℝ}^3,$$ and to the constraint $$|\overline{m}|=1\text{on}{\Omega }_{\epsilon },$$ where W is any continuous function satisfying p-growth assumptions with p> 1. Partial results are also obtained in the case p=1, under an additional assumption on W.

Similar to many mathematical fields also the topic of mathematical programming has its origin in applied problems. But, in contrast to other branches of mathematics, we don't have to dig too deeply into the past centuries to find their roots. The historical tree of mathematical programming, starting from its conceptual roots to its present shape, is remarkably short, and to quote Isaak Newton, we can say: "We are standing on the shoulders of giants". The goal of...

A production inventory problem with limited backlogging and with stockouts is described in a discrete time, stochastic optimal control framework with finite horizon. It is proved by dynamic programming methods that an optimal policy is of (s,S)-type. This means that in every period the policy is completely determined by two fixed levels of the stochastic inventory process considered.

In this paper, we prove that the $L^p$ approximants naturally associated to a supremal functional $\Gamma$-converge to it. This yields a lower semicontinuity result for supremal functionals whose supremand satisfy weak coercivity assumptions as well as a generalized Jensen inequality. The existence of minimizers for variational problems involving such functionals (together with a Dirichlet condition) then easily follows. In the scalar case we show the existence of at least one absolute minimizer (i.e. local...

In this paper, we use $\Gamma$-convergence techniques to study the following variational problem$$S_{\epsilon }^F\left(\Omega \right):=sup\left\{{\epsilon }^{-2^{*}}{\int }_{\Omega }F\left(u\right)dx:{\int }_{\Omega }{\left|\nabla u\right|}^{2}dx\le {\epsilon }^2,u=0\mathrm{on}\partial \Omega \right\},$$where $0\le F\left(t\right)\le {\left|t\right|}^{2^{*}}$, with $2^{*}=\frac{2n}{n-2}$, and $\Omega$ is a bounded domain of ${ℝ}^n$, $n\ge 3$. We obtain a $\Gamma$-convergence result, on which one can easily read the usual concentration phenomena arising in critical growth problems. We extend the result to a non-homogeneous version of problem $S_{\epsilon }^F\left(\Omega \right)$. Finally, a second order expansion in $\Gamma$-convergence permits to identify the concentration points of the maximizing sequences, also in some non-homogeneous case.

Given an open, bounded and connected set $\mathrm{\Omega }\subset {\mathbb{R}}^n$ with Lipschitz boundary and volume $\left|\mathrm{\Omega }\right|$, we prove that the sequence ${\mathcal{F}}_k$ of Dirichlet functionals defined on $H^1\left(\mathrm{\Omega };{\mathbb{R}}^d\right)$, with volume constraints $v^k$ on $m\ge 2$ fixed level-sets, and such that ${\sum }_{i=1}^m{v}_i^k<\left|\mathrm{\Omega }\right|$ for all $k$, $\mathrm{\Gamma }$-converges, as $v^k\to v$ with ${\sum }_{i=1}^m{v}_i^k=\left|\mathrm{\Omega }\right|$, to the squared total variation on $BV\left(V;{\mathbb{R}}^d\right)$, with $v$ as volume constraint on the same level-sets.

The numerical approximation of the minimum problem: ${min}_{A\subseteq \mathrm{\Omega }}\tilde{\mathcal{F}}\left(A\right)$, is considered, where $\tilde{\mathcal{F}}\left(A\right)=P_{\mathrm{\Omega }}\left(A\right)+\cos \left(\theta \right){\mathcal{H}}^{n-1}\left(\partial A\cap \partial \mathrm{\Omega }\right)-{\int }_A\kappa$. The solution to this problem is a set $A\subseteq \mathrm{\Omega }\subset {\mathbb{R}}^n$ with prescribed mean curvature $\kappa$ and contact angle $\theta$ at the intersection of $\partial A$ with $\partial \mathrm{\Omega }$. The functional $\tilde{\mathcal{F}}$ is first relaxed with a sequence of nonconvex functionals defined in $H^1\left(\mathrm{\Omega }\right)$ which, in turn, are discretized by finite elements. The $\mathrm{\Gamma }$-convergence of the discrete functionals to $\tilde{\mathcal{F}}$ as well as the compactness of any sequence of discrete absolute minimizers are proven.

In this note we give a nonstandard characterization of multiple topological $\mathrm{\Gamma }$ operators as sup-min of standard part map.

The notion of $\mathrm{\Gamma }$-limit is extended from the case of functions with values in $\overline{𝐑}$ to the case of those with values in an arbitrary complete lattice and the problem of convergence of Pareto minima related to a convex cone is considered.

In this paper, we prove that the Lp approximants naturally associated to a supremal functional Γ-converge to it. This yields a lower semicontinuity result for supremal functionals whose supremand satisfy weak coercivity assumptions as well as a generalized Jensen inequality. The existence of minimizers for variational problems involving such functionals (together with a Dirichlet condition) then easily follows. In the scalar case we show the existence of at least one absolute minimizer (i.e. local solution)...