We study a class of functions which differ essentially from those which are the sum of a convex function and a regular one and which have interesting properties related to $\mathrm{\Gamma }$-convergence and to problems with non-convex constraints. In particular some results are given for the associated evolution equations.

La semiconcavità è una nozione che generalizza quella di concavità conservandone la maggior parte delle proprietà ma permettendo di ampliarne le applicazioni. Questa è una rassegna dei punti più salienti della teoria delle funzioni semiconcave, con particolare riguardo allo studio dei loro insiemi singolari. Come applicazione, si discuterà una formula di rappresentazione per la soluzione di un modello dinamico per la materia granulare.

The author proved in 2018 that if $G$ is an open subset of a Hilbert space, $f_1,f_2:G\to ℝ$ continuous functions and $\omega$ a nontrivial modulus such that $f_1\le f_2$, $f_1$ is locally semiconvex with modulus $\omega$ and $f_2$ is locally semiconcave with modulus $\omega$, then there exists $f\in C_{\text{loc}}^{1,\omega }\left(G\right)$ such that $f_1\le f\le f_2$. This is a generalization of Ilmanen’s lemma (which deals with linear modulus and functions on an open subset of ${ℝ}^n$). Here we extend the mentioned result from Hilbert spaces to some superreflexive spaces, in particular to $L^p$ spaces, $p\in [2,\infty )$. We also prove...

The aim of this paper is to prove that the relaxation of the elastic-perfectly plastic energy (of a solid made of a Hencky material) is the lower semicontinuous regularization of the plastic energy. We find the integral representation of a non-locally coercive functional. In part II, we will show that the set of solutions of the relaxed problem is equal to the set of solutions of the relaxed problem proposed by Suquet. Moreover, we will prove the existence theorem for the limit analysis problem.

The aim of this paper is to prove that the relaxation of the elastic-perfectly plastic energy (of a solid made of a Hencky material) is the lower semicontinuous regularization of the plastic energy. We find the integral representation of a non-locally coercive functional. We show that the set of solutions of the relaxed problem is equal to the set of solutions of the relaxed problem proposed by Suquet. Moreover, we prove an existence theorem for the limit analysis problem.

The aim of this paper is to prove that the relaxation of the elastic-perfectly plastic energy (of a solid made of a Hencky material with the Signorini constraints on the boundary) is the weak* lower semicontinuous regularization of the plastic energy. We consider an elastic-plastic solid endowed with the von Mises (or Tresca) yield condition. Moreover, we show that the set of solutions of the relaxed problem is equal to the set of solutions of the relaxed problem proposed by Suquet. We deduce that...

We introduce the space $GBD$ of generalized functions of bounded deformation and study the structure properties of these functions: the rectiability and the slicing properties of their jump sets, and the existence of their approximate symmetric gradients. We conclude by proving a compactness results for $GBD$, which leads to a compactness result for the space $GSBD$ of generalized special functions of bounded deformation. The latter is connected to the existence of solutions to a weak formulation of some variational...

We investigate the convergence behavior of the family of double sine integrals of the form ${\int }_0^{\infty }{\int }_0^{\infty }f(x,y)sinuxsinvydxdy$, where (u,v) ∈ ℝ²₊:= ℝ₊ × ℝ₊, ℝ₊:= (0,∞), and f: ℝ²₊ → ℂ is a locally absolutely continuous function satisfying certain generalized monotonicity conditions. We give sufficient conditions for the uniform convergence of the remainder integrals ${\int }_{a₁}^{b₁}{\int }_{a₂}^{b₂}$ to zero in (u,v) ∈ ℝ²₊ as maxa₁,a₂ → ∞ and $b_j>a_j\ge 0$, j = 1,2 (called uniform convergence in the regular sense). This implies the uniform convergence of the partial integrals...

Significant information about the topology of a bounded domain $\Omega$ of a Riemannian manifold $M$ is encoded into the properties of the distance, $d_{\partial \Omega }$, from the boundary of $\Omega$. We discuss recent results showing the invariance of the singular set of the distance function with respect to the generalized gradient flow of $d_{\partial \Omega }$, as well as applications to homotopy equivalence.

There are many types of midconvexities, for example Jensen convexity, t-convexity, (s,t)-convexity. We provide a uniform framework for all the above mentioned midconvexities by considering a generalized middle-point map on an abstract space X. We show that we can define and study the basic convexity properties in this setting.

We prove that for a normed linear space $X$, if $f_1:X\to ℝ$ is continuous and semiconvex with modulus $\omega$, $f_2:X\to ℝ$ is continuous and semiconcave with modulus $\omega$ and $f_1\le f_2$, then there exists $f\in C^{1,\omega }\left(X\right)$ such that $f_1\le f\le f_2$. Using this result we prove a generalization of Ilmanen lemma (which deals with the case $\omega \left(t\right)=t$) to the case of an arbitrary nontrivial modulus $\omega$. This generalization (where a $C_{loc}^{1,\omega }$ function is inserted) gives a positive answer to a problem formulated by A. Fathi and M. Zavidovique in 2010.