Let $\mathrm{\Omega }$ be a bounded open convex set of class $C^2$. Let $a\left(x,H\left(u\right)\right)$ be a non linear operator satisfying the condition (A) (elliptic) with constants $\alpha$, $\gamma$, $\delta$. We prove that a number $\lambda \ge 0$ is an eigenvalue for the operator $a\left(x,H\left(u\right)\right)$ if and only if the number $\alpha \lambda$ is an eigen-value for the operator $\mathrm{\Delta }u$. If $\lambda \ge 0$ , the two systems $a\left(x,H\left(u\right)\right)=\lambda u$ and $\mathrm{\Delta }u=\alpha \lambda u$ have the same solutions. In particular, also the eventual eigen-values of the operator $a\left(x,H\left(u\right)\right)$ should all be negative. Finally, we obtain a sufficient condition for the existence of solutions $u\in H^2\cap H_0^1\left(\mathrm{\Omega }\right)$...

The surjectivity of the operator $D_2+i{x}_2^{2k}{D}_1$ from the Gevrey space ${\mathcal{E}}^{\left\{s\right\}}\left({\mathbb{R}}^2\right)$, $s\ge 1$, onto itself and its non-surjectivity from ${\mathcal{E}}^{\left\{s\right\}}\left({\mathbb{R}}^3\right)$ to ${\mathcal{E}}^{\left\{s\right\}}\left({\mathbb{R}}^3\right)$ is proved.

Let $A$ be a closed set of $M\simeq {\mathbb{R}}^n$, whose conormai cones $x+y_x^{*}\left(A\right)$, $x\in A$, have locally empty intersection. We first show in §1 that $\text{dist}\left(x,A\right)$, $x\in M\setminus A$ is a $C^1$ function. We then represent the n microfunctions of ${\mathcal{C}}_{A|X}$, $X\simeq {\mathbb{C}}^n$, using cohomology groups of ${\mathcal{O}}_X$ of degree 1. By the results of § 1-3, we are able to prove in §4 that the sections of ${\left.{\mathcal{C}}_{A|X}\right|}_{{\dot{\pi }}^{-1}\left(x_0\right)}$, $x_0\in \partial A$, satisfy the principle of the analytic continuation in the complex integral manifolds of ${\left\{H\left({\varphi }_i^C\right)\right\}}_{i=1,\mathrm{\dots },m}$, $\left\{{\varphi }_i\right\}$ being a base for the linear hull of ${\gamma }_{x_0}^{*}\left(A\right)$ in $T_{x_0}^{*}M$; in particular we get ${\left.{\mathrm{\Gamma }}_{A\times {}_{M}T{}^{*}{}_{M}X}\left({\mathcal{C}}_{A|X}\right)\right|}_{\partial A\times {}_M\dot{T}{}^{*}{}_{M}X}=0$. When $A$is a half space with...

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. ...

For a real number $q>1$ and a positive integer $m$, let $Y_m\left(q\right):=\left\{{\sum }_{i=0}^n{ϵ}_i{q}^i:{ϵ}_i\in \left\{0,\pm 1,...,\pm m\right\},n=0,1,...\right\}$. In this paper, we show that $Y_m\left(q\right)$ is dense in $ℝ$ if and only if $q<m+1$ and $q$ is not a Pisot number. This completes several previous results and answers an open question raised by Erdös, Joó and Komornik [8].