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Discrete & Continuous Dynamical Systems - B

2009 , Volume 11 , Issue 1

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Preface
Bernard Dacorogna, Elvira Mascolo and Carlo Sbordone
2009, 11(1): i-ii doi: 10.3934/dcdsb.2009.11.1i +[Abstract](299) +[PDF](29.8KB)
Abstract:
This special issue of Discrete and Continuous Dynamical Systems Series B serves as proceedings of a conference on the calculus of variations and partial differential equations organized in Cortona in May 2007. The organizers were B. Dacorogna (Ecole Polytechnique Fedérale of Lausanne), E. Mascolo (University of Florence) and C. Sbordone (University of Naples) and it was supported by INdAM, Istituto Nazionale di Alta Matematica Francesco Severi.
This conference was the occasion of a special tribute to Paolo Marcellini for his sixtieth birthday.
The themes discussed in the conference were: existence, regularity and comparison for minimizers of the calculus of variations and for solutions of nonlinear elliptic equations and systems, study of nonlinear parabolic equations and systems, applications of variational methods to Euler equation, existence and selection principles for implicit equations, isoperimetric inequalities, applications of partial differential equations to nonlinear elasticity. Let us now see in more details all the contributions.

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Variational models for incompressible Euler equations
Luigi Ambrosio
2009, 11(1): 1-10 doi: 10.3934/dcdsb.2009.11.1 +[Abstract](456) +[PDF](177.0KB)
Abstract:
In this paper we illustrate some recent work [1], [2] on Brenier's variational models for incompressible Euler equations. These models give rise to a relaxation of the Arnold distance in the space of measure-preserving maps and, more generally, measure-preserving plans. We analyze the properties of the relaxed distance, we show a close link between the Lagrangian and the Eulerian model, and we derive necessary and sufficient optimality conditions for minimizers. These conditions take into account a modified Lagrangian induced by the pressure field.
Fastness and continuous dependence in front propagation in Fisher-KPP equations
Margarita Arias, Juan Campos and Cristina Marcelli
2009, 11(1): 11-30 doi: 10.3934/dcdsb.2009.11.11 +[Abstract](384) +[PDF](262.3KB)
Abstract:
We investigate the continuous dependence of the minimal speed of propagation and the profile of the corresponding travelling wave solution of Fisher-type reaction-diffusion equations

$\vartheta_t = (D(\vartheta)\vartheta_x)_x + f(\vartheta)$

with respect to both the reaction term $f$ and the diffusivity $D$.
We also introduce and discuss the concept of fast heteroclinic in this context, which allows to interpret the appearance of sharp heteroclinic in the case of degenerate diffusivity ($D(0)=0)$.

A nonlinear interpolation result with application to the summability of minima of some integral functionals
Lucio Boccardo and Daniela Giachetti
2009, 11(1): 31-42 doi: 10.3934/dcdsb.2009.11.31 +[Abstract](367) +[PDF](179.0KB)
Abstract:
We shall prove results asserting the (global) $L^s$-summability of the minima of integral functionals, using the classical structural assumptions. A feature of the method is that it depends not so much on the minimization problem but rather on the "control from below'' of the structural assumptions. Then the proof concerning the summability of the minima of integral functionals can be easily adapted in order to prove the summability of solutions of nonlinear elliptic equations (even when they are not Euler equations of functionals).
Higher integrability for minimizers of anisotropic functionals
Menita Carozza, Gioconda Moscariello and Antonia Passarelli
2009, 11(1): 43-55 doi: 10.3934/dcdsb.2009.11.43 +[Abstract](468) +[PDF](180.9KB)
Abstract:
In this paper we establish higher integrability results for local minimizers of variational integrals satisfying a degenerate ellipticity condition. The function which measures the degeneracy of the problem is assumed to be exponentially integrable.
Comparison results without strict convexity
Arrigo Cellina, Carlo Mariconda and Giulia Treu
2009, 11(1): 57-65 doi: 10.3934/dcdsb.2009.11.57 +[Abstract](364) +[PDF](145.6KB)
Abstract:
In this paper we establish a comparison result for solutions to the problem

$\mbox{minimize}\int_\Omega l(||\nabla u(x)||)dx $

or to the problem

$\mbox{minimize}\int_\Omega l(\gamma_C(\nabla u(x))dx, $

for a special class of solutions, without assuming neither smoothness nor strict convexity of $l$.

Regularity under sharp anisotropic general growth conditions
Giovanni Cupini, Paolo Marcellini and Elvira Mascolo
2009, 11(1): 67-86 doi: 10.3934/dcdsb.2009.11.67 +[Abstract](384) +[PDF](273.3KB)
Abstract:
We prove boundedness of minimizers of energy-functionals, for instance of the anisotropic type (1) below, under sharp assumptions on the exponents $p_{i}$ in terms of $\overline{p}*$: the Sobolev conjugate exponent of $\overline{p}$; i.e., $\overline{p}*$ = {n\overline{p}}/{n-\overline{p}}, $ $ 1 / \overline{p}$= $\frac{1}{n} \sum_{i=1}^{n}\frac{1}{p_{i}}$. As a consequence, by mean of regularity results due to Lieberman [21], we obtain the local Lipschitz-continuity of minimizers under sharp assumptions on the exponents of anisotropic growth.
Regularity and selecting principles for implicit ordinary differential equations
Bernard Dacorogna and Alessandro Ferriero
2009, 11(1): 87-101 doi: 10.3934/dcdsb.2009.11.87 +[Abstract](454) +[PDF](204.7KB)
Abstract:
Implicit Ordinary or Partial Differential Equations have been widely studied in recent times, essentially from the existence of solutions point of view. One of the main issues is to select a meaningful solution among the infinitely many ones. The most celebrated principle is the viscosity method. This selection principle is well adapted to convex Hamiltonians, but it is not always applicable to the non-convex setting.
In this work we present an alternative selecting principle that singles out the most regular solutions (which do not always coincide with the viscosity ones). Our method is based on a general regularity theorem for Implicit ODEs. We also provide several examples.
A note on the convex hull of sets of finite perimeter in the plane
Alessandro Ferriero and Nicola Fusco
2009, 11(1): 103-108 doi: 10.3934/dcdsb.2009.11.103 +[Abstract](495) +[PDF](116.5KB)
Abstract:
The aim of this paper is to study the minimal perimeter problem for sets containing a fixed set $E$ in $\R^2$ in a very general setting, and to give the explicit solution.
Discrete dynamics of complex bodies with substructural dissipation: Variational integrators and convergence
Matteo Focardi and Paolo Maria Mariano
2009, 11(1): 109-130 doi: 10.3934/dcdsb.2009.11.109 +[Abstract](540) +[PDF](299.1KB)
Abstract:
For the linearized setting of the dynamics of complex bodies we construct variational integrators and prove their convergence by making use of BV estimates on the rate fields. We allow for peculiar substructural inertia and internal dissipation, all accounted for by a d'Alembert-Lagrange-type principle.
Nondivergence elliptic equations with unbounded coefficients
Luigi Greco, Gioconda Moscariello and Teresa Radice
2009, 11(1): 131-143 doi: 10.3934/dcdsb.2009.11.131 +[Abstract](446) +[PDF](208.5KB)
Abstract:
We study the nonvariational equation

$ \sum_{i,j=1}^n a_{ij}(x)\,\frac{\partial^2 u}{\partial x_i\,\partial x_j}=f$

in domains of $r^n$. We assume that the coefficients $a_{ij}$ are in $BMO$ and the equation is elliptic, but not uniformly, and consider $f$ in $L^2(r^n)$, or even in the Zygmund class $L^2\log^\alpha L(r^n)$. We also solve Dirichlet problem.

$L^p$-integrability & weak type $L^{2}$-estimates for the gradient of harmonic mappings of $\mathbb D$
Tadeusz Iwaniec, Gaven Martin and Carlo Sbordone
2009, 11(1): 145-152 doi: 10.3934/dcdsb.2009.11.145 +[Abstract](437) +[PDF](139.4KB)
Abstract:
We give an elementary proof for the integrability properties of the gradient of the harmonic extension of a self homeomorphism of the circle giving explicit bounds for the $p$-norms, $p<2$, estimates in Orlicz classes and also an $L^2(\mathbb D)$-weak type estimate.
The Janossy effect and hybrid variational principles
David Kinderlehrer and Michał Kowalczyk
2009, 11(1): 153-176 doi: 10.3934/dcdsb.2009.11.153 +[Abstract](412) +[PDF](290.5KB)
Abstract:
Light can change the orientation of a liquid crystal. This is the optical Freedericksz transition, discovered by Saupe. In the Janossy effect, the threshold intensity for the optical Freedericksz transition is dramatically reduced by the additon of a small amount of dye to the sample. This has been interpreted as an optically pumped orientational rachet mechanism, similar to the rachet mechanism in biological molecular motors. To interpret the evolution system proposed for this effect requires an innovative gradient flow. Here we introduce this gradient flow and illustrate how it also provides the boundary conditions, some unusual coupling conditions, between the liquid crystal and the dye. An existence theorem for the evolution problem follows as well. Furthermore, we consider the time independent problem and show its local asymptotic stability. Finally we progress toward showing that the proposed model correctly predicts the onset of the Janossy effect.
Higher integrability results for non smooth parabolic systems: The subquadratic case
Chiara Leone, Anna Verde and Giovanni Pisante
2009, 11(1): 177-190 doi: 10.3934/dcdsb.2009.11.177 +[Abstract](476) +[PDF](213.3KB)
Abstract:
In this paper we deal with the study of some regularity properties of weak solutions to non-linear, second-order parabolic systems of the type

$ u_{t}-$div$A(Du)=0, $ $ (x,t)\in \Omega \times (0,T)=\Omega_{T}, $

where $\Omega \subset \mathbb{R}^{n}$ is a bounded domain, $T>0$, $A:\mathbb{R}^{nN}\to \mathbb{R}^{N}$ satisfies a $p$-growth condition and $u:\Omega_{T}\to \mathbb{R}^{N}$. In particular we focus on the case $\frac{2n}{n+2} < p < 2.$

Existence of bounded solutions to some nonlinear degenerate elliptic systems
Francesco Leonetti and Pier Vincenzo Petricca
2009, 11(1): 191-203 doi: 10.3934/dcdsb.2009.11.191 +[Abstract](372) +[PDF](185.3KB)
Abstract:
We prove existence of bounded weak solutions $u: \Omega \subset \R^{n} \to \R^{N}$ for the Dirichlet problem

-div $( a(x, u(x), Du(x) ) ) = f(x),$ $ x \in \Omega$;
$u(x) = 0, $ $ x \in \partial\Omega$

where $\Omega$ is a bounded open set, $a$ is a suitable degenerate elliptic operator and $f$ has enough integrability.

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