$\Gamma$-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...

We prove that the higher integrability of the data $f,f_0$ improves on the integrability of minimizers $u$ of functionals $ℱ$, whose model is $${\int }_{\Omega }\left[{\left|Du\right|}^p+{(det\left(Du\right))}^2-\langle f,Du\rangle +\langle f_0,u\rangle \right]dx,$$ where $u:\Omega \subset {ℝ}^n\to {ℝ}^n$ and $p\ge 2$.

We study the realization $A$ of the operator $\mathcal{A}=\frac{1}{2}\mathrm{△}-(DU,D\cdot )$ in $L^2\left(\mathrm{\Omega },\mu \right)$, where $\mathrm{\Omega }$ is a possibly unbounded convex open set in ${\mathbb{R}}^N$, $U$ is a convex unbounded function such that ${lim}_{x\to \partial \mathrm{\Omega },x\in \mathrm{\Omega }}U\left(x\right)=+\mathrm{\infty }$ and ${lim}_{\left|x\right|\to +\mathrm{\infty },x\in \mathrm{\Omega }}U\left(x\right)=+\mathrm{\infty }$, $DU\left(x\right)$ is the element with minimal norm in the subdifferential of $U$ at $x$, and $\mu \left(dx\right)=c\exp \left(-2U\left(x\right)\right)dx$ is a probability measure, infinitesimally invariant for $\mathcal{A}$. We show that $A$, with domain $D\left(A\right)=\left\{u\in H^2\left(\mathrm{\Omega },\mu \right):\left(DU,Du\right)\in L^2\left(\mathrm{\Omega },\mu \right)\right\}$ is a dissipative self-adjoint operator in $L^2\left(\mathrm{\Omega },\mu \right)$. Note that the functions in the domain of $A$ do not satisfy any particular boundary condition. Log-Sobolev and Poincaré inequalities...

If $\Omega \subset {ℝ}^n$ is a bounded domain with Lipschitz boundary $\partial \Omega$ and $\Gamma$ is an open subset of $\partial \Omega$, we prove that the following inequality $${\left({\int }_{\Omega }{\left|A\left(x\right)\nabla u\left(x\right)\right|}^{p}dx\right)}^{1/p}+{\left({\int }_{\Gamma }{\left|u\left(x\right)\right|}^{p}d{\mathscr{H}}^{n-1}\left(x\right)\right)}^{1/p}\ge c{\parallel u\parallel }_{W^{1,p}\left(\Omega \right)}$$ holds for all $u\in W^{1,p}(\Omega ;{ℝ}^m)$ and $1<p<\infty$, where $${\left(A\left(x\right)\nabla u\left(x\right)\right)}_k=\sum _{i=1}^m\sum _{j=1}^n{a}_k^{ij}\left(x\right)\frac{\partial u_i}{\partial x_j}\left(x\right)(k=1,2,...,r;r\ge m)$$ defines an elliptic differential operator of first order with continuous coefficients on $\overline{\Omega }$. As a special case we obtain $${\int }_{\Omega }{\left|\nabla u\left(x\right)F\left(x\right)+{\left(\nabla u\left(x\right)F\left(x\right)\right)}^T\right|}^{p}dx\ge c{\int }_{\Omega }{\left|\nabla u\left(x\right)\right|}^{p}dx,(*)$$ for all $u\in W^{1,p}(\Omega ;{ℝ}^n)$ vanishing on $\Gamma$, where $F:\overline{\Omega }\to M^{n\times n}\left(ℝ\right)$ is a continuous mapping with $detF\left(x\right)\ge \mu >0$. Next we show that $(*)$ is not valid if $n\ge 3$, $F\in L^{\infty }\left(\Omega \right)$ and $detF\left(x\right)=1$, but does hold if $p=2$, $\Gamma =\partial \Omega$ and $F\left(x\right)$ is symmetric and positive definite in $\Omega$.