Conditions under which the inf-convolution of f and g $f\square g\left(x\right):=in{f}_{y+z=x}(f\left(y\right)+g\left(z\right))$ has the cancellation property (i.e. f □ h ≡ g □ h implies f ≡ g) are treated in a convex analysis framework. In particular, we show that the set of strictly convex lower semicontinuous functions $f:X\to ℝ\cup +\infty$ on a reflexive Banach space such that $li{m}_{\parallel x\parallel \to \infty }f\left(x\right)/\parallel x\parallel =\infty$ constitutes a semigroup, with inf-convolution as multiplication, which can be embedded in the group of its quotients.

We prove the existence of an optimal transport map for the Monge problem in a convex bounded subset of ${ℝ}^d$ under the assumptions that the first marginal is absolutely continuous with respect to the Lebesgue measure and that the cost is given by a strictly convex norm. We propose a new approach which does not use disintegration of measures.

We establish two new formulations of the membrane problem by working in the space of $W_{{\Gamma }_0}^{1,p}(\Omega ,{𝐑}^3)$-Young measures and $W_{{\Gamma }_0}^{1,p}(\Omega ,{𝐑}^3)$-varifolds. The energy functional related to these formulations is obtained as a limit of the $3d$ formulation of the behavior of a thin layer for a suitable variational convergence associated with the narrow convergence of Young measures and with some weak convergence of varifolds. The interest of the first formulation is to encode the oscillation informations on the gradients minimizing sequences...

We establish two new formulations of the membrane problem by working in the space of $W_{{\Gamma }_0}^{1,p}(\Omega ,{𝐑}^3)$-Young measures and $W_{{\Gamma }_0}^{1,p}(\Omega ,{𝐑}^3)$-varifolds. The energy functional related to these formulations is obtained as a limit of the 3d formulation of the behavior of a thin layer for a suitable variational convergence associated with the narrow convergence of Young measures and with some weak convergence of varifolds. The interest of the first formulation is to encode the oscillation informations on the gradients minimizing...

$P_h${ll -div (|Duh|p-2 Duh)=g, & in D Eh uhH1,p0(D Eh). . where $2\le p\le n$ and $E_h$ are random subsets of a bounded open set $D$ of ${\mathbb{R}}^n$$\left(n\ge 2\right)$. By...

The aim of this paper is to study the unilateral contact condition (Signorini problem) for polyconvex functionals with linear growth at infinity. We find the lower semicontinuous relaxation of the original functional (defined over a subset of the space of bounded variations BV(Ω)) and we prove the existence theorem. Moreover, we discuss the Winkler unilateral contact condition. As an application, we show a few examples of elastic-plastic potentials for finite displacements.

To find nonlinear minimization problems are considered and standard $C^2$-regularity assumptions on the criterion function and constrained functions are reduced to $C^{1,1}$-regularity. With the aid of the generalized second order directional derivative for $C^{1,1}$ real-valued functions, a new second order necessary optimality condition and a new second order sufficient optimality condition for these problems are derived.

Continuing earlier work by Székelyhidi, we describe the topological and geometric structure of so-called T4-configurations which are the most prominent examples of nontrivial rank-one convex hulls. It turns out that the structure of T4-configurations in ${ℝ}^{2\times 2}$ is very rich; in particular, their collection is open as a subset of ${\left({ℝ}^{2\times 2}\right)}^4$. Moreover a previously purely algebraic criterion is given a geometric interpretation. As a consequence, we sketch an improved algorithm to detect T4-configurations. ...

We first prove existence and uniqueness of optimal transportation maps for the Monge’s problem associated to a cost function with a strictly convex constraint in the Euclidean plane ℝ2. The cost function coincides with the Euclidean distance if the displacement y − x belongs to a given strictly convex set, and it is infinite otherwise. Secondly, we give a sufficient condition for existence and uniqueness of optimal transportation maps for the original Monge’s problem in ℝ2. Finally, we get existence...