Following Morrey [14] we associate to any measurable symmetric $2\times 2$ matrix valued function $A\left(x\right)$ such that $$\frac{{\left|\xi \right|}^2}{K}\le \left(A\left(x\right)\xi ,\xi \right)\le K{\left|\xi \right|}^2\text{a.e.}x\in \mathrm{\Omega },\mathrm{\forall }\xi \in {\mathbb{R}}^2,$$ $\mathrm{\Omega }\in {\mathbb{R}}^2$ and to any $u\in W^{1,2}\left(\mathrm{\Omega }\right)$ another symmetric $2\times 2$ matrix valued function $\mathcal{A}=\mathcal{A}\left(A,u\right)$ with $det\mathcal{A}=1$ and satisfying $$\frac{{\left|\xi \right|}^2}{K}\le \left(\mathcal{A}\left(x\right)\xi ,\xi \right)\le K{\left|\xi \right|}^2\text{a.e.}x\in \mathrm{\Omega },\mathrm{\forall }\xi \in {\mathbb{R}}^2,$$ The crucial property of $\mathcal{A}$ is that $\mathcal{A}\nabla u=A\nabla u$, if $\nabla u\ne 0$. We study the properties of $\mathcal{A}$ as a function of $A$ and $u$. In particular, we show that, if $A{}_b\to {}^{G}A$, $u_b\rightharpoonup u$, $\nabla u\ne 0$ and $div{A}_b\nabla u_b=0$ then $\mathcal{A}(A{}_b,u{}_b)\to {}^G\mathcal{A}(A,u)$.

Let $X=X_1,\mathrm{\cdots },X_m$ be a family of bounded Lipschitz continuous vector fields on ${\mathbb{R}}^n$. In this paper we prove that if $E$ is a set of finite $X$-perimeter then his $X$-perimeter is the limit of the $X$-perimeters of a sequence of euclidean polyhedra approximating $E$ in $L^1$-norm. This extends to Carnot-Carathéodory geometry a classical theorem of E. De Giorgi.

In this paper, we consider shape optimization problems for the principal eigenvalues of second order uniformly elliptic operators in bounded domains of ${ℝ}^n$. We first recall the classical Rayleigh-Faber-Krahn problem, that is the minimization of the principal eigenvalue of the Dirichlet Laplacian in a domain with fixed Lebesgue measure. We then consider the case of the Laplacian with a bounded drift, that is the operator $-\Delta +v·\nabla$, for which the minimization problem is still well posed. Next, we...

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 $\Omega$.