The Perona–Malik nonlinear parabolic problem, which is widely used in image processing, is investigated in this paper from the numerical point of view. An explicit finite volume numerical scheme for this problem is presented and consistency property is proved.

For a smooth curve $\Gamma$ and a set $\Lambda$ in the plane ${ℝ}^2$, let $AC(\Gamma ;\Lambda )$ be the space of finite Borel measures in the plane supported on $\Gamma$, absolutely continuous with respect to the arc length and whose Fourier transform vanishes on $\Lambda$. Following [12], we say that $(\Gamma ,\Lambda )$ is a Heisenberg uniqueness pair if $AC(\Gamma ;\Lambda )=\left\{0\right\}$. In the context of a hyperbola $\Gamma$, the study of Heisenberg uniqueness pairs is the same as looking for uniqueness sets $\Lambda$ of a collection of solutions to the Klein-Gordon equation. In this work, we mainly address the...

We prove that Perron's method and the method of half-relaxed limits of Barles-Perthame works for the so called B-continuous viscosity solutions of a large class of fully nonlinear unbounded partial differential equations in Hilbert spaces. Perron's method extends the existence of B-continuous viscosity solutions to many new equations that are not of Bellman type. The method of half-relaxed limits allows limiting operations with viscosity solutions without any a priori estimates. Possible applications...

We consider the problem$$\left\{\begin{array}{cc}-\Delta u=u^{\frac{N+2}{N-2}+\lambda }\hfill & \text{in}\Omega \setminus \epsilon \omega ,\hfill \\ u\>0\hfill & \text{in}\Omega \setminus \epsilon \omega ,\hfill \\ u=0\hfill & \text{on}\partial \left(\Omega \setminus \epsilon \omega \right),\hfill \end{array}\right.$$where $\Omega$ and $\omega$ are smooth bounded domains in ${ℝ}^N$, $N\ge 3$, $\epsilon \>0$ and $\lambda \in ℝ.$ We prove that if the size of the hole $\epsilon$ goes to zero and if, simultaneously, the parameter $\lambda$ goes to zero at the appropriate rate, then the problem has a solution which blows up at the origin.

On prouve que le problème de Cauchy local pour l’équation d’onde sur-critique dans ${ℝ}^d$, $\square u+u^p=0$, $p$ impair, avec $d\ge 3$ et $p\>(d+2)/(d-2)$, est mal posé dans $H^{\sigma }$ pour tout $\sigma \in ]1,{\sigma }_{\mathrm{crit}}[$, où ${\sigma }_{\mathrm{crit}}=d/2-2/(p-1)$ est l’exposant critique.

L'objet de cet exposé est l'étude d'équations d'évolution de type parabolique, périodiques, que l'on pénalise par un terme linéaire, antisymétrique. Par application des méthodes de S. Schochet pour le cas hyperbolique, on obtient un développement asymptotique des solutions de telles équations. La méthode suivie consiste à étudier l'influence de fortes oscillations en temps dans des systèmes paraboliques. Cette théorie est appliquée à deux systèmes décrivant le comportement de fluides géophysiques,...

This paper deals with linear partial differential-algebraic equations (PDAEs) which have a hyperbolic part. If the spatial differential operator satisfies a Gårding-type inequality in a suitable function space setting, a perturbation index can be defined. Theoretical and practical examples are considered.