arXiv subjects
Publications and source records attributed to Andrea Torricelli.
We consider integral functionals with fast growth and the lagrangian explicitly depending on $u$. We prove that the local minimizers are locally Lipschitz continuous.
We study the simultaneous homogenization and dimension reduction of an energy functional with linear growth defined on the space of manifold valued Sobolev functions. The study is carried out by $\Gamma$-convergence, providing an integral representation result in the space of manifold constrained functions with bounded variation
We study simultaneous homogenization and dimensional reduction of integral functionals for maps in manifold-valued Sobolev spaces. Due to the superlinear growth regime, we prove that the density of the $\Gamma$-limit is a tangential quasiconvex integrand represented by a cell formula.
We prove a homogenization result in terms of two-scale Young measures for non-local integral functionals. The result is obtained by means of a characterization of two-scale Young measures.
We prove some regularity results for a priori bounded local minimizers of non-autonomous integral functionals of the form $$\mathcal{F}(v,Ω)=\int_ΩF(x,Dv)dx,$$ under the constraint $v \ge ψ$ a.e. in $Ω$, where $ψ$ is a fixed obstacle function. Assuming that the coefficients of the partial map $x \mapsto D_ξF(x,ξ)$ satisfy a suitable Sobolev regularity, we are able to obtain higher differentiability and Lipschitz continuity results for the local minimizers.
We study minimizers of non-autonomous energies with minimal growth and coercivity assumptions on the energy. We show that the minimizer is nevertheless the solution of the relevant Euler--Lagrange equation or inequality. The main tool is an extension result for convex $C^1$-energies.
We provide necessary and sufficient conditions on the density $W:\mathbb R^d\times\mathbb R ^d\to\mathbb R$ in order to ensure the sequential weak* lower semicontinuity of the functional $J: W^{1,\infty}(I;\mathbb R^d)\to \mathbb R$, defined as \begin{align*} J(u):=ess\,sup_{I\times I}W(u'(x), u'(y)), \end{align*} when $I$ is an open and bounded interval of $\mathbb R$. We also show that, when $d=1$, the lower semicontinuous envelope of $I$ in general can be obtained by replacing $W$ by its separately level convex envelope.
Let us consider the autonomous obstacle problem \begin{equation*} \min_v \int_ΩF(Dv(x)) \, dx \end{equation*} on a specific class of admissible functions, where we suppose the Lagrangian satisfies proper hypotheses of convexity and superlinearity at infinity. Our aim is to characterize the solution, which exists and it is unique, thanks to a primal-dual formulation of the problem. The proof is based on classical arguments of Convex Analysis and on Calculus of Variations' techniques.
In this paper we consider a class of obstacle problems of the type %\begin{equation*} %\int_Ω\left \, \dx\ge0\qquad\forall %φ\in W^{1,q}(Ω) \quad {\mathrm{s.t.}} \quad φ\ge ψ%\end{equation*} \begin{equation*} \min \left\{\int_Ωf(x, Dv)\, \dx\,:\, v\in \mathcal{K}_ψ(Ω)\right\} \end{equation*} where $ψ$ is the obstacle, $\mathcal{K}_ψ(Ω)=\{v\in u_0+W^{1, p}_{0}(Ω, \R): v\geψ\text{ a.e. in }Ω\}$, with $u_0 \in W^{1,p}(Ω)$ a fixed boundary datum, the class of the admissible functions and the integrand $f(x, Dv)$ satisfies non standard $(p,q)$-growth conditions. \\ We prove higher differentiability results for bounded solutions of the obstacle problem under dimension-free conditions on the gap between the growth and the ellipticity exponents. Moreover, also the Sobolev assumption on the partial map $x\mapsto A(x, ξ)$ is independent of the dimension $n$ and this, in some cases, allows us to manage coefficients in a Sobolev class below the critical one $W^{1,n}$.
We study the asymptotic behavior of a family of functionals which penalize a short-range interaction of convolution type between a finite perimeter set and its complement. We first compute the pointwise limit and we obtain a lower estimate on more regulars sets. Finally, some examples are discussed.