In this text, we present two recent results on the characterization of the lack of compactness of some critical Sobolev embedding. The first one derived in [5] deals with an abstract framework including Sobolev, Besov, Triebel-Lizorkin, Lorentz, Hölder and BMO spaces. The second one established in [3] concerns the lack of compactness of $H^1\left({ℝ}^2\right)$ into the Orlicz space. Although the two results are expressed in the same manner (by means of defect measures) and rely on the defect of compactness due to concentration...

This paper is aimed at the description of the multi-dimensional finite volume solver EULER, which has been developed for the numerical solution of the compressible Euler equations during several last years. The present overview of numerical schemes and the explanation of numerical techniques and tricks which have been used for EULER could be of certain interest not only for registered users but also for numerical mathematicians who have decided to implement a finite volume solver themselves. This...

We consider a reaction-diffusion system of the activator-inhibitor type with unilateral boundary conditions leading to a quasivariational inequality. We show that there exists a positive eigenvalue of the problem and we obtain an instability of the trivial solution also in some area of parameters where the trivial solution of the same system with Dirichlet and Neumann boundary conditions is stable. Theorems are proved using the method of a jump in the Leray-Schauder degree.

On présente ici une approche directe et géométrique pour le calcul des déterminants d’opérateurs de type Schrödinger sur un graphe fini. Du calcul de l’intégrale de Fresnel associée, on déduit le déterminant. Le calcul des intégrales de Fresnel est grandement facilité par l’utilisation simultanée du théorème de Fubini et d’une version linéaire du calcul symbolique des opérateurs intégraux de Fourier. On obtient de façon directe une formule générale exprimant le déterminant en terme des conditions...

This paper investigates the inverse problem of finding the time-dependent diffusion coefficient in a quasilinear parabolic equation with the nonlocal boundary and integral overdetermination conditions. Under some natural regularity and consistency conditions on the input data the existence, uniqueness and continuously dependence upon the data of the solution are shown. Finally, some numerical experiments are presented.

The problem of determining the source term $F(x,t)$ in the linear parabolic equation $u_t={\left(k\left(x\right)u_x(x,t)\right)}_x+F(x,t)$ from the measured data at the final time $u(x,T)=\mu \left(x\right)$ is formulated. It is proved that the Fréchet derivative of the cost functional $J\left(F\right)=\parallel {\mu }_T{\left(x\right)-u(x,T)\parallel }_0^2$ can be formulated via the solution of the adjoint parabolic problem. Lipschitz continuity of the gradient is proved. An existence result for a quasi solution of the considered inverse problem is proved. A monotone iteration scheme is obtained based on the gradient method. Convergence rate is proved....

For a bounded and sufficiently smooth domain $\mathrm{\Omega }$ in ${\mathbb{R}}^N$, $N\ge 2$, let ${\left({\lambda }_k\right)}_{k=1}^{\mathrm{\infty }}$ and ${\left({\phi }_k\right)}_{k=1}^{\mathrm{\infty }}$ be respectively the eigenvalues and the corresponding eigenfunctions of the problem (with Neumann boundary conditions) $$-\text{div}\left(a\left(x\right)\nabla {\phi }_k\right)+q\left(x\right){\phi }_k={\lambda }_k\varrho \left(x\right){\phi }_k\text{ in }\mathrm{\Omega },a\frac{\partial }{\partial \mathbf{n}}{\phi }_k=0\text{ su }\partial \mathrm{\Omega }.$$ We prove that knowledge of the Dirichlet boundary spectral data ${\left({\lambda }_k\right)}_{k=1}^{\mathrm{\infty }}$, ${\left({\phi }_{k|\partial \mathrm{\Omega }}\right)}_{k=1}^{\mathrm{\infty }}$ determines uniquely the Neumann-to-Dirichlet (or the Steklov- Poincaré) map $\gamma$ for a related elliptic problem. Under suitable hypothesis on the coefficients $a,q,\varrho$ their identifiability is then proved. We prove also analogous results for Dirichlet...

In the present paper, we consider nonlinear optimal control problems with constraints on the state of the system. We are interested in the characterization of the value function without any controllability assumption. In the unconstrained case, it is possible to derive a characterization of the value function by means of a Hamilton-Jacobi-Bellman (HJB) equation. This equation expresses the behavior of the value function along the trajectories arriving or starting from any position x. In the constrained...