Abstract
In this paper, we first provide counterexamples showing that sublevels of prox-regular functions and levels of differentiable mappings with Lipschitz derivatives may fail to be prox-regular. Then, we prove the uniform prox-regularity of such sets under usual verifiable qualification conditions. The preservation of uniform prox-regularity of intersection and inverse image under usual qualification conditions is also established. Applications to constrained optimization problems are given.

Abstract
In this paper, we study the well−posedness (in the sense of existence and uniqueness of a solution) of a discontinuous sweeping process involving prox-regular sets in Hilbert spaces. The variation of the moving set is controlled by a positive Radon measure and the perturbation is assumed to satisfy a Lipschitz property. The existence of a solution with bounded variation is achieved thanks to the Moreau’s catching-up algorithm adapted to this kind of problem. Various properties and estimates of jumps of the solution are also provided. We give sufficient conditions to ensure the uniform prox- regularity when the moving set is described by inequality constraints. As an application, we consider a nonlinear differential complementarity system which is a combination of an ordinary differential equation with a nonlinear complementarily condition. Such problems appear in many areas such as nonsmooth mechanics, nonregular electrical circuits and control systems.

Abstract
In this paper, we study the existence of solutions for a variant of discontinuous Moreau sweeping process in the infinite dimensional setting. The sets involved are assumed to be uniformly prox-regular and move with bounded variation. The sweeping process is perturbed by a sum of Lipschitz continu- ous single-valued mapping and a scalarly upper semicontinuous multimapping satisfying a linear growth condition with respect to a compact set.

Abstract
In this paper, we study the existence of solutions for a time and state-dependent discontinuous nonconvex second order sweeping process with a multivalued perturbation. The moving set is assumed to be prox-regular, relatively ball-compact with a bounded variation. The perturbation of the normal cone is a scalarly upper semi- continuous convex valued multimapping satisfying a linear growth condition possibly time-dependent. As an application of the theoretical results, we investigate the theory of evolution quasi-variational inequalities.

Abstract
In this paper, we first investigate the prox-regularity behaviour of solution mappings to generalized equations. This study is realized through a nonconvex uniform Robinson−Ursescu type theorem. Then, we derive new significant results for the preservation of prox-regularity under various and usual set operations. The role and applications of prox-regularity of solution sets of generalized equations are illustrated with dynamical systems with constraints.

Abstract
The paper surveys in great generality several fundamental contributions in the literature on regularization of sweeping processes under the control of the moving set via the Hausdorff-Pompeiu distance. In addition, a large complete new study is provided for the regularization of prox-regular sweeping processes in the significantly weaker situation when merely a suitable truncated Hausdorff distance is involved for the control of the moving set.

Abstract
In this paper, we deal with the existence of solutions for perturbed state-dependent Moreau’s sweeping processes. Two ways are investigated to realize such a study, depending on the nature of the used scheme, namely implicit or semi-implicit. In both cases, our evolution problem is described in a general Hilbert space by a prox-regular moving set controlled through the truncated Hausdorff-Pompeiu distance. The normal cone involved is perturbed by a sum of a single-valued mapping and a multimapping.

Abstract
This paper is devoted to the existence and uniqueness of solutions for perturbed sweeping process measure differential inclusions in infinite dimensional setting. The possibly unbounded moving set is prox-regular and controlled only through the truncated Hausdorff-Pompeiu distance. The normal cone involved is perturbed by a kind of Carathéodory mapping satisfying a time-dependent hypomonotonicity assumption on bounded sets. Various properties of the solution mapping are also provided.

Abstract
This paper is devoted to a new family of measure differential inclusions in Hilbert spaces. We show that it encompasses the first order BV prox-regular sweeping process and the second order one with outward normal at the velocity. Through a new suitable mixed catching-up algorithm coming from first and second order sweeping process theory, we provide sufficient conditions ensuring the existence of a trajectory solution for our evolution problem.

Abstract
In this paper, we present diverse new metric properties that prox-regular sets shared with convex ones. At the heart of our work lie the Legendre- Fenchel transform and complements of balls. First, we show that a connected prox-regular set is completely determined by the Legendre-Fenchel transform of a suitable perturbation of its indicator function. Then, we prove that such a function is also the right tool to extend, to the context of prox-regular sets, the famous connection between the distance function and the support function of a convex set. On the other hand, given a prox-regular set, we examine the intersection of complements of open balls containing the set. We establish that the distance of a point to a prox-regular set is the maximum of the distances of the point from boundaries of all such complements separating the set and the point. This is in the line of the known result expressing the distance from a convex set in terms of separating hyperplanes. To the best of our knowledge, these results are new in the literature and show that the class of prox-regular sets have good properties known in convex analysis.

Abstract
This paper is devoted to a differential inclusion encompassing in a general Hilbert space first order state-dependent sweeping process and second order one with outward normal at the velocity. Our evolution problem is governed by the normal cone of a bounded subsmooth set which moves in an absolute continuous way. The existence of a trajectory solution is established through an appropriate Moreau’s catching-up algorithm.

Abstract
We consider a new class of inclusions in Hilbert spaces for which we provide an existence and uniqueness result. The proof is based on arguments of monotonicity, convexity and fixed point. We use this result to establish the unique solvability of an associated class of Moreau’s sweeping processes. Next, we give two applications in Solid Mechanics. The first one concerns the study of a time-dependent constitutive law with unilateral constraints and memory term. The second one is related to a frictional contact problem for viscoelastic materials. For both problems we describe the physical setting, list the assumptions on the data and provide existence and uniqueness results.

Abstract
In this paper, we address the problem of maximizing the Steklov eigenvalues with a diameter constraint. We provide an estimate of the Steklov eigenvalues for a convex domain in terms of its diameter and volume, and we show the existence of an optimal convex domain. We establish that balls are never maximizers, even for the first nontrivial eigenvalue that contrasts with the case of volume or perimeter constraints. Under an additional regularity assumption, we are able to prove that the Steklov eigenvalue is multiple for the optimal domain. We illustrate our theoretical results by giving some optimal domains in the plane thanks to a numerical algorithm.

Abstract
We consider a special type of sweeping process in real Hilbert spaces, governed by two (possibly history-dependent) operators. We associate to this problem an auxiliary time-dependent inclusion for which we establish an existence and uniqueness result. The proof is based on arguments of convex analysis and fixed point theory. From the unique solvability of the intermediate inclusion, we derive the existence of a unique solution to the considered sweeping processes. Our theoretical results find various applications in contact mechanics. As an example, we consider a frictional contact problem for viscoelastic materials. We list the assumptions on the data and provide a variational formulation of the problem, in a form of a sweeping process for the strain field. Then, we prove the unique solvability of the sweeping process and use it to obtain the existence of a unique weak solution to the viscoelastic contact problem.

Abstract
We consider an abstract inclusion in a real Hilbert space, governed by an almost history-dependent operator and a time-dependent multimapping with prox-regular values. We establish the unique solvability of the inclusion, under appropriate assumptions on the data. The proof is based on arguments of monotonicity, fixed point and the prox-regularity property. We then use our result in order to deduce some direct consequences, including an existence and uniqueness result for a class of sweeping processes associated to prox-regular sets. Finally, we provide an example in finite dimensional case, inspired by a rheological model in Solid Mechanics.

Abstract
In this paper, we provide in a general Hilbert space new characterizations of uniform prox- regularity involving outside but sufficiently close points of considered sets. We show that the complement of a prox-regular set is nothing but the union of closed balls with common radius. We derive from this that the prox-regularity of a given closed set is equivalent to the semiconvexity property of its distance function. Various estimates involving the metric projection mapping to a prox-regular set are also established.

Abstract
This paper is devoted to nonconvex/prox-regular separations of sets in Hilbert spaces. We introduce the Legendre-Fenchel r-conjugate of a prescribed function and r-quadratic support functionals and points of a given set, all associated to a positive constant r. By means of these concepts we obtain nonlinear functional separations for points and prox-regular sets. In addition to such functional separations, we also establish geometric separation results with balls for a prox-regular set and a strongly convex set.

Abstract
The aim of the present paper is twofold. On one hand, we show that the strong convexity of a set is equivalent to the semiconcavity of its associated farthest distance function. On the other hand, we establish that the farthest distance of a point from a strongly convex set is the minimum of farthest distances of the given point from suitable closed balls separating the set and the point. Various other results on strongly convex sets are also provided.

Abstract
We introduce in a general Hilbert space the class of ρ(·)-strongly convex sets. Various characterizations and properties of such sets involving the farthest distance function and the farthest points are provided.

Abstract
In this paper, we establish through an openness condition the metric subregularity of a multimapping with normal ω(·)-regularity of either the graph or values. Various preservation results for prox-regular and subsmooth sets are also provided.

Abstract
This paper provides some new results for the classes of prox-regular sets and strongly convex sets with variable radii in a general Hilbert space.

Abstract
In this paper, we consider a general Mayer optimal control problem whose dynamics is defined regionally, and, additionally, we suppose that the interface between two regions is nonsmooth in the sense that it is described by a locally Lipschitz continuous function. Our objective is to derive a hybrid maximum principle in this setting. Doing so, we consider a sequence of mollifiers which allows us to approximate uniformly the interface between two regions by a sequence of smooth functions. This makes possible to apply the hybrid maximum principle on a sequence of approximated optimal control problems involving the smooth interface in place of the nonsmooth one. By passing to the limit as the approximation parameter tends to zero, we obtain the desired necessary optimality conditions in the form of a nonsmooth hybrid maximum principle. Our approach relies on the obtention of subgradients of a locally Lipschitz continuous function via mollifiers.