We study the effect on the solutions of linear parabolic equations of Gaussian symmetrization. This kind of symmetrization, which transforms a domain into an half-space with the same Gaussian measure, is specifically useful when the domain is unbounded. The main tools we use are the properties of the weighted rearrangements and in particular the isoperimetric inequality with respect to Gaussian measure.

Strongly elliptic differential operators with (possibly) unbounded lower-order coefficients are shown to generate $C_0$ semigroups on $L^p(\mathbb R^N)$, $1 < p < +\infty$. An explicit characterization of the domain is given.

In this paper, starting from the bi-fluid Euler-Maxwell system, we derive a complete set of Zakharov-type equations describing laser-plasma interactions. This system involves a quasilinear part which is not hyperbolic and exhibits some elliptic zones. This difficulty is overcome by making a change of unknowns that are strongly related to the dispersive part. This change of variable is a symmetrization of the quasilinear part and is the key of this paper. This shows that the Cauchy problem is locally well-posed.

We investigate mathematically a minimization problem issued from computational chemistry, and known as the Optimal Effective Potentiel (OEP) problem. We propose a weak formulation of the problem, that we show to be well-posed in a simple case, and we address various related questions.

We prove the existence of at least two solutions for a fourth-order equation, which includes the vortex equations for the $U(1)$ and $CP(1)$ self-dual Maxwell-Chern-Simons models as special cases. Our method is variational, and it relies on an ``asymptotic maximum principle" property for a special class of supersolutions to this fourth-order equation.

We analyze the stability of positive solutions to systems of the form \[ \begin{cases} - \Delta u_{i} = f_{i}(u_{1},u_{2},\dots,u_{m}) & \quad \mbox{ in }\ \Omega \\ u_{i} = 0 & \quad \mbox{ on }\ \partial \Omega \end{cases} \] where $ \Omega $ is a bounded region in $ {\mathbb R}^{n}\, (n \geq 1) $ with smooth boundary $ \partial \Omega $, and $ f_{i} : [0,\infty)^m \rightarrow {\mathbb R} $ are $ C^{1} $ functions for $ i=1,\dots, m $. In particular, we establish conditions for stability/instability when the system is cooperative and strictly coupled ($ \frac{\partial f_{i}}{\partial u_{j}} \geq 0, \ i \neq j,\ \sum_{j=1,j \neq i}^m(\frac{\partial f_i}{\partial u_j})^2 > 0$). When $ m=2, $ we extend this analysis for strictly coupled competitive systems ($ \frac{\partial f_{i}}{\partial u_{j}} < 0, \ i \neq j $). We apply our results to various examples, each one of different characteristics, and further analyze systems with unequal diffusion coefficients.

In this paper, we discuss the fundamental linear theory for a large class of delay differential equations with impulses. We show, using the general theory of integrated semigroups, that we can associate an integrated semigroup with any delay differential equation with impulses, and we have determined the infinitesimal generator of this integrated semigroup.

In this paper we study periodic, nonlinear, second-order differential inclusions, driven by the differential operator $$ x\rightarrow (\alpha(x)\|x'\|^{p-2}x')' $$ and involving a maximal monotone term $A$ and a multivalued nonlinearity $F(t,x)$ which satisfies the Hartman condition. We do not assume that $domA$ is all of $\mathbb{R}^{N}$, and so our formulation incorporates variational inequalities. Then we obtain partial generalizations. First, we allow $F$ to depend on $x'$, but for $p=2$ and for the scalar problem ($N=1$). Second, we assume a general multivalued, nonlinear differential operator $x\rightarrow \alpha(x,x')'$; the nonlinearity $F$ depends also on $x'$, but the boundary conditions are Dirichlet. Our methods are based on notions and techniques from multivalued analysis and from the theory of operators of monotone type.