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

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$\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 ${\Omega}_{\epsilon}=\omega \times (-\epsilon ,\epsilon )$, $\omega \subset {\mathbb{R}}^{2}$, whose energy is given by$${\mathcal{E}}_{\epsilon}\left(\overline{m}\right)=\frac{1}{\epsilon}{\int}_{{\Omega}_{\epsilon}}\left(W(\overline{m},\nabla \overline{m})+\frac{1}{2}\nabla \overline{u}\xb7\overline{m}\right)\phantom{\rule{0.166667em}{0ex}}\mathrm{d}x$$subject to$$\text{div}(-\nabla \overline{u}+\overline{m}{\chi}_{{\Omega}_{\epsilon}})=0\phantom{\rule{1.0em}{0ex}}\text{on}{\mathbb{R}}^{3},$$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 ${\Omega}_{\epsilon}=\omega \times (-\epsilon ,\epsilon )$, $\omega \subset {\mathbb{R}}^{2}$, whose energy is given by $${\mathcal{E}}_{\epsilon}\left(\overline{m}\right)=\frac{1}{\epsilon}{\int}_{{\Omega}_{\epsilon}}\left(W(\overline{m},\nabla \overline{m})+\frac{1}{2}\nabla \overline{u}\xb7\overline{m}\right)\phantom{\rule{0.166667em}{0ex}}\mathrm{d}x$$ subject to $$\text{div}(-\nabla \overline{u}+\overline{m}{\chi}_{{\Omega}_{\epsilon}})=0\phantom{\rule{1.0em}{0ex}}\phantom{\rule{4.0pt}{0ex}}\text{on}\phantom{\rule{4.0pt}{0ex}}{\mathbb{R}}^{3},$$ and to the constraint $$|\overline{m}|=1\phantom{\rule{4.0pt}{0ex}}\text{on}\phantom{\rule{4.0pt}{0ex}}{\Omega}_{\epsilon},$$ 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.

This article is the starting point of a series of works whose aim is the study of deterministic control problems where the dynamic and the running cost can be completely different in two (or more) complementary domains of the space ℝN. As a consequence, the dynamic and running cost present discontinuities at the boundary of these domains and this is the main difficulty of this type of problems. We address these questions by using a Bellman approach: our aim is to investigate how to define properly...

The present paper concerns the problem of the flow through a semipermeable membrane of infinite thickness. The semipermeability boundary conditions are first considered to be monotone; these relations are therefore derived by convex superpotentials being in general nondifferentiable and nonfinite, and lead via a suitable application of the saddlepoint technique to the formulation of a multivalued boundary integral equation. The latter is equivalent to a boundary minimization problem with a small...

In this paper, we employ the reduced basis method as a surrogate model for the solution of linear-quadratic optimal control problems governed by parametrized elliptic partial differential equations. We present a posteriori error estimation and dual procedures that provide rigorous bounds for the error in several quantities of interest: the optimal control, the cost functional, and general linear output functionals of the control, state, and adjoint variables. We show that, based on the assumption...

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