arXiv subjects
Publications and source records attributed to Michela Eleuteri.
We consider a class of anisotropic energy integrals including fast and slow growth under a unified approach, proving that the local minimizers are locally Lipschitz continuous.
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 $Γ$-limit is a tangential quasiconvex integrand represented by a cell formula.
The aim of this paper is to deal with the asymptotics of generalized Orlicz norms when the lower growth rate tends to infinity. $Γ$-convergence results and related representation theorems in terms of $L^\infty$ functionals are proven for sequences of generalized Orlicz energies under mild convexity assumptions. This latter hypothesis is removed in the variable exponent setting.
We propose a homogenized supremal functional rigorously derived via $L^p$-approximation by functionals of the type $\underset{x\inΩ}{\mbox{ess-sup}}\hspace{0.03cm} f\left(\frac{x}{\varepsilon}, Du\right)$, when $Ω$ is a bounded open set of $\mathbb R^n$ and $u\in W^{1,\infty}(Ω;\mathbb R^d)$. The homogenized functional is also deduced directly in the case where the sublevel sets of $f(x,\cdot)$ satisfy suitable convexity properties, as a corollary of homogenization results dealing with pointwise gradient constrained integral functionals.
We establish the Lipschitz regularity of the a priori bounded local minimizers of integral functionals with non autonomous energy densities satisfying non standard growth conditions under a sharp bound on the gap between the growth and the ellipticity exponent.
We consider integral functionals with slow growth and explicit dependence on u of the lagrangian; this includes many relevant examples, as, for instance, in elastoplastic torsion problems or in image restoration problems. Our aim is to prove that the local minimizers are locally Lipschitz continuous. The proof makes use of recent results concerning the Bounded Slope Conditions.
$3d-2d$ dimensional reduction for hyperelastic thin films modeled through energies with point dependent growth, assuming that the sample is clamped on the lateral boundary, is performed in the framework of $Γ$-convergence. Integral representation results, with a more regular lagrangian related to the original energy density, are provided for the lower dimensional limiting energy, in different contexts.
Motivated by the image denoising problem and the undesirable stair-casing effect of the total variation method, we introduce bounded variation spaces with generalized Orlicz growth. Our setup covers earlier variable exponent and double phase models. We study the norm and modular of the new space and derive a formula for the modular in terms of the Lebesgue decomposition of the derivative measure and a location dependent recession function. We also show that the modular can be obtained as the $Γ$-limit of uniformly convex approximating energies.
In this paper we prove the Lipschitz regularity for local minimizers of convex variational integrals of the form \[ \mathfrak{F}( v, Ω)= \int_Ω \! F(x, Dv(x)) \, dx, \] where, for ${n > 2}$ and $N\ge 1$, $Ω$ is a bounded open set in $\mathbb{R}^n$, $u \in W^{1,1}(Ω, \mathbb{R}^N)$ and the energy density $F:Ω\times \mathbb{R}^{N \times n}\to \mathbb{R}$ satisfies the so called variable growth conditions. The main novelty of the paper is that we assume an almost critical regularity in the Orlicz Sobolev setting for the energy density as a function of the $x$ variable.
A way to measure the lower growth rate of $φ:Ω\times [0,\infty) \to [0,\infty)$ is to require $t \mapsto φ(x,t)t^{-r}$ to be increasing in $(0,\infty)$. If this condition holds with $r=1$, then \[ \inf_{u\in f+W^{1, φ}_0(Ω)}\int_Ωφ(x, |\nabla u|) \, dx \] with boundary values $f\in W^{1,φ}(Ω)$ does not necessary have a minimizer. However, if $φ$ is replaced by $φ^p$, then the growth condition holds with $r=p > 1$ and thus (under some additional conditions) the corresponding energy integral has a minimizer. We show that a sequence $(u_p)$ of such minimizers convergences when $p \to 1^+$ in a suitable $\mathrm{BV}$-type space involving generalized Orlicz growth and obtain the $Γ$-convergence of functionals with fixed boundary values and of functionals with fidelity terms. %We complement our results by showing that some previous papers by some of the authors are included in our analysis.
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.
We consider some energy integrals under slow growth and we prove that the local minimizers are locally Lipschitz continuous. Many examples are given, either with subquadratic $p,q-$growth and/or anisotropic growth.
The aim of the paper is to show that the solutions to variational problems with non-standard growth conditions satisfy a corresponding variational inequality without any smallness assumptions on the gap between growth and coercitivity exponents. Our results rely on techniques based on Convex Analysis that consist in establishing duality formulas and pointwise relations between minimizers and corresponding dual maximizers, for suitable approximating problems, that are preserved passing to the limit. In this respect we are able to show that the right class of competitors are the functions with finite energy, in agreement with the unconstrained results.
In a recent paper [5] a smooth function f : [0; 1] --> R with all derivatives vanishing at 0 has been considered and a global condition, showing that f is indeed identically 0, has been presented. The purpose of this note is to replace the classical Fundamental Theorem of Calculus for the Riemann integral, as it has been used in [5], with a weaker form going back to Volterra [7], which is little known. Therefore the proof we propose in this paper turns to be important also from the teaching point of view, as long as in literature there are very few examples in which explicitly the lower integral and the upper integral of a function appear (usually the assumption that the function is Riemann-integrable is required).
We study the $Γ$-convergence of the functionals $F_n(u):= || f(\cdot,u(\cdot),Du(\cdot))||_{p_n(\cdot)}$ and $\mathcal{F}_n(u):= \int_Ω \frac{1}{p_n(x)} f^{p_n(x)}(x,u(x),Du(x))dx$ defined on $X\in \{L^1(Ω,\mathbb{R}^d), L^\infty(Ω,\mathbb{R}^d), C(Ω,\mathbb{R}^d)\}$ (endowed with their usual norms) with effective domain the Sobolev space $W^{1,p_n(\cdot)}(Ω, \mathbb{R}^d )$. Here $Ω\subseteq \mathbb{R}^N$ is a bounded open set, $N,d \ge 1$ and the measurable functions $p_n: \overlineΩ \rightarrow (1, + \infty) $ satisfy the conditions ${\mathop{\rm ess\: sup }}_{\ \overline Ω} p_n \le \, β\, {\mathop{\rm ess\: inf }}_{\ \overline Ω} p_n $ for a fixed constant $β> 1$ and $ {\mathop{\rm ess\: inf }}_{\ \overline Ω} p_n \rightarrow + \infty$ as $n \rightarrow + \infty$. We show that when $f(x,u,\cdot)$ is level convex and lower semicontinuous and it satisfies a uniform growth condition from below, then, as $n\to \infty$, the sequences $(F_n)_n$ $Γ$-converges in $X$ to the functional $F$ represented as $F(u)= || f(\cdot,u(\cdot),Du(\cdot))||_{\infty}$ on the effective domain $W^{1,\infty}(Ω, \mathbb{R}^d )$. Moreover we show that the $Γ$-$\lim_n \mathcal F_n$ is given by the functional $ \mathcal{F}(u):=\left\{\begin {array}{lll} \!\!\!\!\!\! & 0 & \hbox{if } || f(\cdot,u(\cdot),Du(\cdot)) ||_{\infty}\leq 1,\\ \!\!\!\!\!\! & +\infty & \hbox{otherwise in } X.\\ \end{array}\right. $
In this paper we prove the local boundedness as well as the local Lipschitz continuity for solutions to a class of obstacle problems of the type $$\min\left\{\int_Ω{F(x, Dz)}: z\in \mathcal{K}_ψ(Ω)\right\}.$$ Here $\mathcal{K}_ψ(Ω)$ is set of admissible functions $z \in W^{1,p}(Ω)$ such that $z \ge ψ$ a.e. in $Ω$, $ψ$ being the obstacle and $Ω$ being an open bounded set of $\mathbb{R}^n$, $n \ge 2$. The main novelty here is that we are assuming $ F(x, Dz)$ satisfying $(p,q)$-growth conditions {and less restrictive assumptions on the obstacle with respect to the existing regularity results}.
In this note we give a proof of the Sobolev and Morrey embedding theorems based on the representation of functions in terms of the fundamental solution of suitable partial differential operators. We also prove the compactness of the Sobolev embedding. We first describe this method in the classical setting, where the fundamental solution of the Laplace equation is used, to recover the classical Sobolev and Morrey theorems. We next consider degenerate Kolmogorov equations. In this case, the fundamental solution is invariant with respect to a non-Euclidean translation group and the usual convolution is replaced by an operation that is defined in accordance with this geometry. We recover some known embedding results and we prove the compactness of the Sobolev embedding. We finally apply our regularity results to a kinetic equation.
We consider weak solutions of degenerate second order partial differential equations of Kolmogorov-Fokker-Planck type with measurable coefficients in divergence form. We give a geometric statement of the Harnack inequality recently proven by Golse, Imbert, Mouhot and Vasseur. As a corollary we obtain a strong maximum principle.