In this short paper, we are concerned with the stability of nonlinear bilevel programs. A stability problem is proven and an example is given to illustrate this theorem.

The computation of glacier movements leads to a system of nonlinear partial differential equations. The existence and uniqueness of a weak solution is established by using the calculus of variations. A discretization by the finite element method is done. The solution of the discrete problem is proved to be convergent to the exact solution. A first simple numerical algorithm is proposed and its convergence numerically studied.

The aim of this paper is to study a quasistatic unilateral contact problem between an elastic body and a foundation. The constitutive law is nonlinear and the contact is modelled with a normal compliance condition associated to a unilateral constraint and Coulomb's friction law. The adhesion between contact surfaces is taken into account and is modelled with a surface variable, the bonding field, whose evolution is described by a first-order differential equation. We establish a variational formulation...

We consider the integral functional $J\left(u\right)={\int }_{\Omega }\left[f\right(\left|Du\right|\left)-u\right]dx$, $u\in W_0^{1,1}\left(\Omega \right)$, where $\Omega \subset {ℝ}^n$, $n\ge 2$, is a nonempty bounded connected open subset of ${ℝ}^n$ with smooth boundary, and $ℝ\ni s\mapsto f\left(\right|s\left|\right)$ is a convex, differentiable function. We prove that if $J$ admits a minimizer in $W_0^{1,1}\left(\Omega \right)$ depending only on the distance from the boundary of $\Omega$, then $\Omega$ must be a ball.

The aim of this article is to propose a new method for the grey-level image classification problem. We first present the classical variational approach without and with a regularization term in order to smooth the contours of the classified image. Then we present the general topological asymptotic analysis, and we finally introduce its application to the grey-level image classification problem.

In this paper we analyse the structure of approximate solutions to the compatible two well problem with the constraint that the surface energy of the solution is less than some fixed constant. We prove a quantitative estimate that can be seen as a two well analogue of the Liouville theorem of Friesecke James Müller. Let $H=\left(\begin{array}{cc}\sigma & 0\\ 0& {\sigma }^{-1}\end{array}\right)$ for $\sigma \>0$. Let $0\<{\zeta }_1\<1\<{\zeta }_2\<\infty$. Let $K:=SO\left(2\right)\cup SO\left(2\right)H$. Let $u\in W^{2,1}\left(Q_1\left(0\right)\right)$ be a ${\mathrm{C}}^1$ invertible bilipschitz function with $\mathrm{Lip}\left(u\right)\<{\zeta }_2$, $\mathrm{Lip}\left(u^{-1}\right)\<{\zeta }_1^{-1}$. There exists positive constants ${𝔠}_1\<1$ and ${𝔠}_2\>1$ depending only on $\sigma$, ${\zeta }_1$, ${\zeta }_2$ such that if $ϵ\in \left(0,{𝔠}_1\right)$ and $u$ satisfies the...