Nous étudions les résonances de Rayleigh créées par un obstacle strictement convexe à bord analytique en dimension 2. Nous montrons qu’il existe exactement deux suites de résonances $\left(z_{k,+}\right)$ et $\left(z_{k,-}\right)$ convergeant exponentiellement vite vers l’axe réel dans un voisinage polynomial de l’axe réel, et exponentiellement proches d’une suite de quasimodes réels. De plus, $k^{-1}\Re z_{k,\pm }$ est un symbole analytique d’ordre 0 en la variable $k^{-1}$ dont on donne le premier terme du développement. Nous construisons pour cela des quasimodes...

We deal with the boundary value problem $$\begin{array}{ccccccc}\hfill -\Delta u\left(x\right)& ={\lambda }_{1}u\left(x\right)+g\left(\nabla u\left(x\right)\right)+h\left(x\right),\hfill & & x\in \Omega u\left(x\right)\hfill & \hfill =0,& & \hfill x\in \partial \Omega \end{array}$$ where $\Omega \subset {ℝ}^N$ is an smooth bounded domain, ${\lambda }_1$ is the first eigenvalue of the Laplace operator with homogeneous Dirichlet boundary conditions on $\Omega$, $h\in L^{max\{2,N/2\}}\left(\Omega \right)$ and $g:{ℝ}^N⟶ℝ$ is bounded and continuous. Bifurcation theory is used as the right framework to show the existence of solution provided that $g$ satisfies certain conditions on the origin and at infinity.

The aim of this paper is to study the existence of variational solutions to a nonhomogeneous elliptic equation involving the $N$-Laplacian $$-{\Delta }_{N}u\equiv -{div\left(\right|\nabla u|}^{N-2}\nabla u)=e(x,u)+h\left(x\right)\text{in}\Omega$$ where $u\in W_0^{1,N}\left({ℝ}^N\right)$, $\Omega$ is a bounded smooth domain in ${ℝ}^N$, $N\ge 2$, $e(x,u)$ is a critical nonlinearity in the sense of the Trudinger-Moser inequality and $h\left(x\right)\in {\left(W_0^{1,N}\right)}^{*}$ is a small perturbation.

We study the spectral projection associated to a barrier-top resonance for the semiclassical Schrödinger operator. First, we prove a resolvent estimate for complex energies close to such a resonance. Using that estimate and an explicit representation of the resonant states, we show that the spectral projection has a semiclassical expansion in integer powers of $h$, and compute its leading term. We use this result to compute the residue of the scattering amplitude at such a resonance. Eventually, we...

In this paper we propose an original approach for the simulation of the time-dependent response of a floating elastic plate using the so-called Singularity Expansion Method. This method consists in computing an asymptotic behaviour for large time obtained by means of the Laplace transform by using the analytic continuation of the resolvent of the problem. This leads to represent the solution as the sum of a discrete superposition of exponentially damped oscillating motions associated to the poles...