Shape optimisation for a class of semilinear variational inequalities with applications to damage models

The present contribution investigates shape optimisation problems for a class of semilinear elliptic variational inequalities with Neumann boundary conditions. Sensitivity estimates and material derivatives are firstly derived in an abstract operator setting where the operators are defined on polyhedral subsets of reflexive Banach spaces. The results are then refined for variational inequalities arising from minimisation problems for certain convex energy functionals considered over upper obstacle sets in $H^1$. One particularity is that we allow for dynamic obstacle functions which may arise from another optimisation problems. We prove a strong convergence property for the material derivative and establish state-shape derivatives under regularity assumptions. Finally, as a concrete application from continuum mechanics, we show how the dynamic obstacle case can be used to treat shape optimisation problems for time-discretised brittle damage models for elastic solids. We derive a necessary optimality system for optimal shapes whose state variables approximate desired damage patterns and/or displacement fields

Free boundary problems involving singular weights

In this paper we initiate the investigation of free boundary minimization problems ruled by general singular operators with $A_2$ weights. We show existence and boundedness of minimizers. The key novelty is a sharp $C^{1+\gamma}$ regularity result for solutions at their singular free boundary points. We also show a corresponding non-degeneracy estimate

Hessian measures II

In our previous paper on this topic, we introduced the notion of k-Hessian measure associated with a continuous k-convex function in a domain \Om in Euclidean n-space, k=1,...,n, and proved a weak continuity result with respect to local uniform convergence. In this paper, we consider k-convex functions, not necessarily continuous, and prove the weak continuity of the associated k-Hessian measure with respect to convergence in measure. The proof depends upon local integral estimates for the gradients of k-convex functions.Comment: 26 pages, published versio

Nonlinear Eigenvalues and Bifurcation Problems for Pucci's Operator

In this paper we extend existing results concerning generalized eigenvalues of Pucci's extremal operators. In the radial case, we also give a complete description of their spectrum, together with an equivalent of Rabinowitz's Global Bifurcation Theorem. This allows us to solve equations involving Pucci's operators

A qualitative mathematical analysis of a class of variational inequalities via semi-complementarity problems: applications in electronics

International audienceThe main object of this paper is to present a general mathematical theory applicable to the study of a large class of linear variational inequalities arising in electronics. Our approach uses recession tools so as to define a new class of problems that we call "semi-complementarity problems". Then we show that the study of semi-complementarity problems can be used to prove new qualitative results applicable to the study of linear variational inequalities of the second kin