Some electromagnetic materials have, in a given frequency range, an effective dielectric permittivity and/or a magnetic permeability which are real-valued negative coefficients when dissipation is neglected. They are usually called metamaterials. We study a scalar transmission problem between a classical dielectric material and a metamaterial, set in an open, bounded subset of Rd, with d = 2,3. Our aim is to characterize occurences where the problem is well-posed within the Fredholm (or coercive...

Some electromagnetic materials have, in a given frequency range, an effective dielectric permittivity and/or a magnetic permeability which are real-valued negative coefficients when dissipation is neglected. They are usually called metamaterials. We study a scalar transmission problem between a classical dielectric material and a metamaterial, set in an open, bounded subset of Rd, with d = 2,3. Our aim is to characterize occurences where the problem is well-posed within the Fredholm (or coercive...

We investigate the existence of solutions of the Dirichlet problem for the differential inclusion $0\in \Delta x\left(y\right)+{\partial }_{x}G(y,x\left(y\right))$ for a.e. y ∈ Ω, which is a generalized Euler-Lagrange equation for the functional $J\left(x\right)={\int }_{\Omega }1/2\left|\nabla x\right(y\left)\right|²-G(y,x(y\left)\right)dy$. We develop a duality theory and formulate the variational principle for this problem. As a consequence of duality, we derive the variational principle for minimizing sequences of J. We consider the case when G is subquadratic at infinity.

In this paper, we are concerned with the asymptotically linear elliptic problem -Δu + λ0u = f(u), u ∈ H01(Ω) in an exterior domain Ω = RnO (N ≥ 3) with O a smooth bounded and star-shaped open set, and limt→+∞ f(t)/t = l, 0 < l < +∞. Using a precise deformation lemma and algebraic topology argument, we prove under our assumptions that the problem possesses at least one positive solution.

In this article we prove for $1<p<\infty$ the existence of the $L^p$-Helmholtz projection in finite cylinders $\Omega$. More precisely, $\Omega$ is considered to be given as the Cartesian product of a cube and a bounded domain $V$ having $C^1$-boundary. Adapting an approach of Farwig (2003), operator-valued Fourier series are used to solve a related partial periodic weak Neumann problem. By reflection techniques the weak Neumann problem in $\Omega$ is solved, which implies existence and a representation of the $L^p$-Helmholtz projection as...

We consider a semilinear elliptic eigenvalues problem on a ball of ${\Re }^n$ and show that all the eigenfunctions and eigenvalues, can be obtained from the Lane-Emden function.

We prove the existence of the least positive eigenvalue with a corresponding nonnegative eigenfunction of the quasilinear eigenvalue problem $$\begin{array}{ccc}\hfill -div\left(a(x,u)\right|{|}^{p-2}\nabla u)=& {\lambda b(x,u)\left|u\right|}^{p-2}u\text{in}\Omega ,u=\hfill & \hfill 0\text{on}\partial \Omega ,\end{array}$$ where $\Omega$ is a bounded domain, $p>1$ is a real number and $a(x,u)$, $b(x,u)$ satisfy appropriate growth conditions. Moreover, the coefficient $a(x,u)$ contains a degeneration or a singularity. We work in a suitable weighted Sobolev space and prove the boundedness of the eigenfunction in $L^{\infty }\left(\Omega \right)$. The main tool is the investigation of the associated homogeneous eigenvalue problem and an application...

The paper is devoted to the description of some connections between the mean curvature in a distributional sense and the mean curvature in a variational sense for several classes of non-smooth sets. We prove the existence of the mean curvature measure of $\partial E$ by using a technique introduced in [4] and based on the concept of variational mean curvature. More precisely we prove that, under suitable assumptions, the mean curvature measure of $\partial E$ is the weak limit (in the sense of distributions) of the mean...

We prove sharp inequalities in weighted Sobolev spaces. Our approach is based on the blow-up technique applied to some nonlinear Neumann problems.

We consider the $p$-Laplacian operator on a domain equipped with a Finsler metric. We recall relevant properties of its first eigenfunction for finite $p$ and investigate the limit problem as $p\to \infty$.

The Picone-type identity for the half-linear second order partial differential equation $$\sum _{i=1}^n\frac{\partial }{\partial x_i}\Phi \left(\frac{\partial u}{\partial x_i}\right)+c\left(x\right)\Phi \left(u\right)=0,\Phi \left(u\right):={\left|u\right|}^{p-2}u,p>1,$$ is established and some applications of this identity are suggested.

Inequalities concerning the integral of |∇u|2 on the subsets where |u(x)| is greater than k can be used in order to prove regularity properties of the function u. This method was introduced by Ennio De Giorgi e Guido Stampacchia for the study of the regularity of the solutions of Dirichlet problems.