Chiara Leone

Chiara Leone
University of Naples Federico II | UNINA · Department of Mathematics and Applications "R.Caccioppoli"

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35
Publications
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340
Citations

Publications

Publications (35)
Preprint
We prove a new $\mathcal{A}$-caloric approximation lemma compatible with an Orlicz setting. With this result, we establish a partial regularity result for parabolic systems of the type $$ u_{t}- {\rm div} \,a(Du)=0. $$ Here the growth of $a$ is bounded by the derivative of an $N$-function $\varphi$. The primary assumption for $\varphi$ is that $t\v...
Article
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We consider a quasilinear degenerate parabolic equation driven by the orthotropic p-Laplacian. We prove that local weak solutions are locally Lipschitz continuous in the spatial variable, uniformly in time.
Preprint
In this paper we prove a H\"older partial regularity result for weak solutions $u:\Omega\to \mathbb{R}^N$, $N\geq 2$, to non-autonomous elliptic systems with general growth of the type: \begin{equation*} -\rm{div}\, a(x, u, Du)= b(x, u, Du) \quad \mbox{ in } \Omega. \end{equation*} The crucial point is that the operator $a$ satisfies very weak regu...
Preprint
Full-text available
We consider a quasilinear degenerate parabolic equation driven by the orthotropic $p-$Laplacian. We prove that local weak solutions are locally Lipschitz continuous in the spatial variable, uniformly in time.
Article
In this paper we prove a partial Hölder regularity result for weak solutions \begin{document}$ u:\Omega\to \mathbb{R}^N $\end{document}, \begin{document}$ N\geq 2 $\end{document}, to non-autonomous elliptic systems with general growth of the type: \begin{document}$ \begin{equation*} -{\rm{div}} a(x, u, Du) = b(x, u, Du) \quad \;{\rm{ in }}\; \Omega...
Article
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In this paper we consider a linear elliptic equation in divergence form 0.1∑i,jDj(aij(x)Diu)=0inΩ.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sum _...
Book
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Queste note sono una versione espansa delle lezioni del corso di calcolo delle variazioni tenute da Carlo Mantegazza al Dipartimento di Matematica e Applicazioni “Renato Caccioppoli” dell’Università Federico II di Napoli. L’esposizione del soggetto, che è quasi esclusivamente relativo ai problemi unidimensionali, segue in un qualche senso lo svilup...
Article
We prove a regularity result for the weak solutions of H-systems in dimensions n≥3.
Article
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We prove that local weak solutions of the orthotropic p-harmonic equation are locally Lipschitz, for every p≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p\ge 2$$\e...
Article
We present a lower semicontinuity result for free discontinuity energies with a quasiconvex volume term having non standard growth and a quite general surface term.
Article
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We prove higher differentiability of bounded local minimizers to some widely degenerate functionals, verifying superquadratic anisotropic growth conditions. In the two dimensional case, we prove that local minimizers to a model functional are locally Lipschitz continuous functions, without any restriction on the anisotropy.
Article
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We prove partial regularity for minimizers of vectorial integrals of the Calculus of Variations, with general growth condition, imposing quasiconvexity assumptions only in an asymptotic sense.
Data
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We prove a lower semicontinuity result for polyconvex functionals of the Calculus of Variations along sequences of maps u : R^n → R^m in W^(1,m), 2 ≤ m ≤ n, bounded in W^(1,m−1) and convergent in L^1 under mild technical conditions but without any extra coercivity assumption on the integrand.
Article
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We prove a lower semicontinuity result for polyconvex functionals of the calculus of variations along sequences of maps u:Ω⊂ℝ n →ℝ m in W 1,m , 2≤m≤n, which are bounded in W 1,m-1 and convergent in L 1 , under mild technical conditions but without any extra coercivity assumption on the integrand.
Article
We study the Hölder regularity of weak solutions to the evolutionary p-Laplacian system with critical growth on the gradient. We establish a natural criterion for proving that a small solution and its gradient are locally Hölder continuous almost everywhere. Actually our regularity result recovers the classical result in the case p=2p=2[16] and can...
Article
We prove the existence of a global “small” weak solution to the flow of the H-system with initial–boundary conditions. We also analyze its time asymptotic behavior. Finally we give a stability result for weak solutions to the heat flow of higher dimensional H-systems.RésuméOn démontre lʼexistence dʼune solution faible globale « petite » du flux du...
Article
Full-text available
We prove a lower semicontinuity result for free discontinuity energies with a quasiconvex volume term having non standard growth and a surface term.
Article
We prove a C 2,α partial regularity result for local minimizers of polyconvex variational integrals of the type I(u)=òW |D2u|2+g(det(D2u))dx{I(u)=\int_\Omega |D^{2}u|^2+g({\det}(D^2u))dx}, where Ω is a bounded open subset of \mathbb R2{ \mathbb {R}^{2}}, u Î Wloc2,2(W){u\in W_{loc}^{2,2}(\Omega)} and g Î C2(\mathbb R){g\in C^{2}(\mathbb {R})} i...
Article
We deal with the study of some regularity properties of weak solutions to nonlinear, second-order parabolic systems of the type u t -divA(Du)=0,(x,t)∈Ω×(0,T)=Ω T , where Ω⊂ℝ n is a bounded domain, T>0, A:ℝ nN →ℝ N satisfies a p-growth condition and u:Ω T →ℝ N . In particular we focus on the case 2n n+2<p<2.
Article
In this paper we deal with the study of regularity properties of weak solutions to nonlinear, second-order parabolic systems of the typeut−divA(Du)=0,(x,t)∈Ω×(−T,0)=ΩT, where Ω⊂Rn is a bounded domain, T>0, A:RnN→RN and u:ΩT→RN. In particular we provide higher fractional differentiability, partial regularity and estimates for the dimension of the si...
Article
We establish a local Lipschitz regularity result for local minimizers of variational integrals under the assumption that the integrand becomes appropriately elliptic at infinity. The exponent that measures the ellipticity of the integrand is assumed to be less than two.
Chapter
The aim of this work is to study obstacle problems associated to monotone operators when the forcing term is a bounded Radon measure. Existence, uniqueness, stability results, and properties of the solutions are investigated.
Article
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In this paper we study the relaxation with respect to the L1 norm of integral functional of the type F(u) = ∫Ω f(x,u, ∇u) dx, u ∈ W1,1(Ω;Sd-1), where Ω is a bounded open set of RN, Sd-1 denotes the unite sphere in Rd, N and d being any positive integers, and f satisfies linear growth conditions in the gradient variable. In analogy with the unconstr...
Article
We prove a semi-continuity theorem for an integral functional made up by a polyconvex energy and a surface term. Our result extends a well-known result by Ball to the BV framework.
Article
We study existence and regularity of distributional solutions for possibly degenerate quasi-linear parabolic problems having a first order term which grows quadratically in the gradient. The model problem we refer to is the following(1) Here Ω is a bounded open set in RN, T>0. The unknown function u=u(x,t) depends on x∈Ω and . The symbol ∇u denotes...
Article
We give a definition for Obstacle Problems with measure data and general obstacles. For such problems we prove existence and uniqueness of solutions and consistency with the classical theory of Variational Inequalities. Continuous dependence with respect to data is discussed.
Article
We deal with minima for convex functionals of the calculus of variations when the forcing term is a function of L 1 (Ω), exhibiting a notion of minimum equivalent to that one given in [L. Boccardo, Ric. Mat. 49, Suppl., 135-154 (2000; Zbl 1009.49002)]. This new formulation allows us to make easier the proofs of existence and uniqueness results alre...
Article
The obstacle problem with measure data associated with a nonlinear elliptic differential operator is analyzed. The theory of variational inequalities is studied in the classical context of data. An approach is proposed, which allows to obtain a stability result, with respect to strong convergence of data.
Article
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Gamma -convergence techniques and relaxation results of constrained energy functionals are used to identify the limiting energy as the thickness epsilon approaches zero of a ferromagnetic thin structure Omega (epsilon) = omega x (epsilon,epsilon), omega subset of R-2, whose energy is given by epsilon (epsilon)((m) over bar) = 1/epsilon integral (Om...
Article
We give a definition for Obstacle Problems with measure data and general obstacles. For such problems we prove existence and uniqueness of solutions and consistency with the classical theory of Variational Inequalities. Continuous dependence with respect to data is discussed.
Article
Full-text available
We study the notion of solution to an obstacle problem for a strongly monotone and Lipschitz operator A, when the forcing term is a bounded Radon measure. We obtain existence and uniqueness results. We study also some properties of the obstacle reactions associated with the solutions of the obstacle problems, obtaining the Lcwy­Stampacchia inequali...
Article
We study the convergence properties of the solutions of some elliptic obstacle problems with measure data, under the simultaneous perturbation of the operator, the forcing term and the obstacle. Ref. S.I.S.S.A. 80/99/M (July 99) Stability results for obstacle problems with measure data 1 1. Introduction Obstacle problems when the data do not belong...
Article
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We study existence and regularity of distributional solutions for possibly degenerate quasi-linear parabolic problems having a first order term which grows quadratically in the gradient. The model problem we refer to is the following  

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Project (1)
Project
Prove higher order regularity for solutions of degenerate/singular equations modeled on the orthotropic p-Laplacian