We shall prove a weak comparison principle for quasilinear elliptic operators $-\mathrm{div}\left(a\right(x,\nabla u\left)\right)$ that includes the negative $p$-Laplace operator, where $a:\Omega \times {ℝ}^N\to {ℝ}^N$ satisfies certain conditions frequently seen in the research of quasilinear elliptic operators. In our result, it is characteristic that functions which are compared belong to different spaces.

The existence of at least one non-decreasing sequence of positive eigenvalues for the problem driven by both $p(·)$-Harmonic and $p(·)$-biharmonic operators $$\begin{array}{c}{\Delta }_{p\left(x\right)}^{2}u-{\Delta }_{p\left(x\right)}u=\lambda w\left(x\right){\left|u\right|}^{q\left(x\right)-2}u\text{in}\Omega ,\\ u\in W^{2,p(·)}\left(\Omega \right)\cap W_0^{1,p(·)}\left(\Omega \right),\end{array}$$ is proved by applying a local minimization and the theory of the generalized Lebesgue-Sobolev spaces $L^{p(·)}\left(\Omega \right)$ and $W^{m,p(·)}\left(\Omega \right)$.

In the theory of elliptic equations, the technique of Schwarz symmetrization is one of the tools used to obtain a priori bounds for classical and weak solutions in terms of general information on the data. A basic result says that, in the absence of lower-order terms, the symmetric rearrangement of the solution $u$ of an elliptic equation, that we write $u^{*}$, can be compared pointwise with the solution of the symmetrized problem. The main question we address here is the modification of the method to...

This paper is devoted to a study of harmonic mappings $\phi$ of a harmonic space $\tilde{E}$ on a harmonic space $E$ which are related to a family of harmonic mappings of $\tilde{E}$ into $\tilde{E}$. In this way balayage in $E$ may be reduced to balayage in $E$. In particular, a subset $A$ of $E$ is polar if and only if ${\phi }^{-1}\left(A\right)$ is polar. Similar result for thinness. These considerations are applied to the heat equation and the Laplace equation.

We study the propagation of electromagnetic waves in a guide the section of which is a thin annulus. Owing to the presence of a small parameter, explicit approximations of the TM and TE eigenmodes are obtained. The cases of smooth and non smooth boundaries are presented.

We study the propagation of electromagnetic waves in a guide the section of which is a thin annulus. Owing to the presence of a small parameter, explicit approximations of the TM and TE eigenmodes are obtained. The cases of smooth and non smooth boundaries are presented.

The starting point of the analysis in this paper is the following situation: “In a bounded domain in ${ℝ}^2$, let a finite set of points be given. A triangulation of that domain has to be found, whose vertices are the given points and which is ‘suitable’ for the linear conforming Finite Element Method (FEM).” The result of this paper is that for the discrete Poisson equation and under some weak additional assumptions, only the use of Delaunay triangulations preserves the maximum principle.

This paper is devoted to a conditional stability estimate related to the ill-posed Cauchy problems for the Laplace's equation in domains with C1,1 boundary. It is an extension of an earlier result of [Phung, ESAIM: COCV9 (2003) 621–635] for domains of class C∞. Our estimate is established by using a Carleman estimate near the boundary in which the exponential weight depends on the distance function to the boundary. Furthermore, we prove that this stability estimate is nearly optimal and induces...

We discuss the stability of "critical" or "equilibrium" shapes of a shape-dependent energy functional. We analyze a problem arising when looking at the positivity of the second derivative in order to prove that a critical shape is an optimal shape. Indeed, often when positivity -or coercivity- holds, it does for a weaker norm than the norm for which the functional is twice differentiable and local optimality cannot be a priori deduced. We solve this problem for a particular but significant example....