A second order minimality condition for the Mumford-Shah functional

A new necessary minimality condition for the Mumford-Shah functional is derived by means of second order variations. It is expressed in terms of a sign condition for a nonlocal quadratic form on $H^1_0(\Gamma)$, $\Gamma$ being a submanifold of the regular part of the discontinuity set of the critical point. Two equivalent formulations are provided: one in terms of the first eigenvalue of a suitable compact operator, the other involving a sort of nonlocal capacity of $\Gamma$. A sufficient condition for minimality is also deduced. Finally, an explicit example is discussed, where a complete characterization of the domains where the second variation is nonnegative can be given.Comment: 30 page

Multiplicity of Positive Solutions for an Obstacle Problem in R

In this paper we establish the existence of two positive solutions for the obstacle problem \displaystyle \int_{\Re}\left[u'(v-u)'+(1+\lambda V(x))u(v-u)\right] \geq \displaystyle \int_{\Re} f(u)(v-u), \forall v\in \Ka where $f$ is a continuous function verifying some technical conditions and \Ka is the convex set given by \Ka =\left\{v\in H^{1}(\Re); v \geq \varphi \right\}, with $\varphi \in H^{1}(\Re)$ having nontrivial positive part with compact support in $\Re$. \vspace{0.2cm} \noindent \emph{2000 Mathematics Subject Classification} : 34B18, 35A15, 46E39. \noindent \emph{Key words}: Obstacle problem, Variational methods, Positive solutions.Comment: To appear in Progress in Nonlinear Differential Equations and their Application