Global Lipschitz continuity for minima of degenerate problems

International audienceWe consider the problem of minimizing the Lagrangian [F (∇u)+f u] among functions on Ω ⊂ R N with given boundary datum ϕ. We prove Lipschitz regularity up to the boundary for solutions of this problem, provided Ω is convex and ϕ satisfies the bounded slope condition. The convex function F is required to satisfy a qualified form of uniform convexity only outside a ball and no growth assumptions are made

Continuity of solutions of a problem in the calculus of variations

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Nonsmooth Lyapunov pairs for differential inclusions governed by operators with nonempty interior domain

The general theory of Lyapunov stability of first-order differential inclusions in Hilbert spaces has been studied by the authors in the previous paper (Adly et al. in Nonlinear Anal 75(3): 985–1008, 2012). This new contribution focuses on the case when the interior of the domain of the maximally monotone operator governing the given differential inclusion is nonempty; this includes in a natural way the finite-dimensional case. The current setting leads to simplified, more explicit criteria and permits some flexibility in the choice of the generalized subdifferentials. Some consequences of the viability of closed sets are given. Our analysis makes use of standard tools from convex and variational analysis. © 2015, Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society