Given a reaction-diffusion system which exhibits Turing's diffusion-driven instability, the influence of unilateral obstacles of opposite sign (source and sink) on bifurcation and critical points is studied. In particular, in some cases it is shown that spatially nonhomogeneous stationary solutions (spatial patterns) bifurcate from a basic spatially homogeneous steady state for an arbitrarily small ratio of diffusions of inhibitor and activator, while a sufficiently large ratio is necessary in the...

In his book on convex polytopes [2] A. D. Aleksandrov raised a general question of finding variational formulations and solutions to geometric problems of existence of convex polytopes in ${ℝ}^{n+1}$, n ≥ 2, with prescribed geometric data. Examples of such problems for closed convex polytopes for which variational solutions are known are the celebrated Minkowski problem [2] and the Gauss curvature problem [20]. In this paper we give a simple variational proof of existence for the A. D. Aleksandrov problem...

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.