# American Institute of Mathematical Sciences

ISSN:
1078-0947

eISSN:
1553-5231

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## Discrete & Continuous Dynamical Systems - A

2011 , Volume 31 , Issue 4

A special issue
in honour of E. De Giorgi and G. Stampacchia

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2011, 31(4): i-vi doi: 10.3934/dcds.2011.31.4i +[Abstract](469) +[PDF](296.2KB)
Abstract:
The International Conference "Variational Analysis and Applications" devoted to the memory of Ennio De Giorgi and Guido Stampacchia was held at the International School of Mathematics in Erice from May 9 to May 17 of 2009. About thirty lecturers from every part of the world took part in the conference and the workshop was enriched by the award of the "Third Gold Medal G. Stampacchia" to the young researcher Camillo de Lellis. Some of the lecturers, together with their well-known students, presented a paper for this special issue in honour of Ennio De Giorgi and Guido Stampacchia.

2011, 31(4): 1017-1021 doi: 10.3934/dcds.2011.31.1017 +[Abstract](589) +[PDF](216.1KB)
Abstract:
$\Gamma$-convergence was introduced by Ennio De Giorgi in a series of papers published between 1975 and 1983. In the same years he developed many applications of this tool to a great variety of asymptotic problems in the calculus of variations and in the theory of partial differential equations.
2011, 31(4): 1023-1038 doi: 10.3934/dcds.2011.31.1023 +[Abstract](616) +[PDF](699.7KB)
Abstract:
Using Lyapunov's stability and LaSalle's invariance principle for nonsmooth dynamical systems, we establish some conditions for finite-time stability of evolution variational inequalities. The theoretical results are illustrated by some examples drawn from electrical circuits involving nonsmooth elements like diodes.
2011, 31(4): 1039-1051 doi: 10.3934/dcds.2011.31.1039 +[Abstract](431) +[PDF](345.3KB)
Abstract:
We study the evolution of a class of quasistatic problems, which describe frictional contact between a body and a foundation. The constitutive law of the materials is elastic, or visco-elastic: with short or long memory, and the contact is modelled by a general subdifferential condition on the velocity. We derive weak formulations for the models and establish existence and uniqueness results. The proofs are based on evolution variational inequalities, in the framework of monotone operators and $fi$xed point methods. We show the approach of the viscoelastic solutions to the corresponding elastic solutions, when the viscosity tends to zero. Finally we also study the approach to short memory visco-elasticity when the long memory relaxation coefficients vanish.
2011, 31(4): 1053-1067 doi: 10.3934/dcds.2011.31.1053 +[Abstract](325) +[PDF](415.5KB)
Abstract:
For $\Omega\subset \mathbb{R}^2$ a bounded open set with $\mathcal{C}^1$ boundary, we study the regularity of the variational solution $v\in W_0^{1,2}(\Omega)$ to the quasilinear elliptic equation of Leray-Lions \begin{equation*} - \,\textrm{div}\, A(x, \nabla v) = f \end{equation*} when $f$ belongs to the Zygmund space $L(\log L)^{\delta}(\Omega)$, $\frac{1}{2} \leq \delta \leq 1$. We prove that $|\nabla v|$ belongs to the Lorentz space $L^{2, 1/\delta}(\Omega)$.
2011, 31(4): 1069-1096 doi: 10.3934/dcds.2011.31.1069 +[Abstract](498) +[PDF](480.4KB)
Abstract:
This paper concerns new developments on first-order necessary conditions in set-valued optimization with applications of the results obtained to deriving refined versions of the so-called second fundamental theorem of welfare economics. It is shown that equilibrium marginal prices at local Pareto-type optimal allocations of nonconvex economies are in fact adjoint elements/ multipliers in necessary conditions for fully localized minimizers of appropriate constrained set-valued optimization problems. The latter notions are new in multiobjective optimization and reduce to conventional notions of minima for scalar problems. Our approach is based on advanced tools of variational analysis and generalized differentiation.
2011, 31(4): 1097-1113 doi: 10.3934/dcds.2011.31.1097 +[Abstract](434) +[PDF](419.5KB)
Abstract:
The weighted traffic equilibrium problem introduced in [17], in which the equilibrium conditions have been expressed in terms of a weighted variational inequality, studies a transportation network in presence of congestion. For such a problem, existence and regularity theorems have been proved in [8]. In this paper, we analyze the dual problem and characterize the weighted traffic equilibrium solutions by means of Lagrange multipliers, which allow to describe the behavior of the weighted transportation network.
2011, 31(4): 1115-1128 doi: 10.3934/dcds.2011.31.1115 +[Abstract](419) +[PDF](605.8KB)
Abstract:
In this paper we analyze the relaxed form of a shape optimization problem with state equation $$$$\left\{\begin{array}{ll} -div\big(a(x)Du\big)=f\qquad\hbox{in }D\\ \hbox{boundary conditions on }\partial D. \end{array} \right.$$$$ The new fact is that the term $f$ is only known up to a random perturbation $\xi(x,\omega)$. The goal is to find an optimal coefficient $a(x)$, fulfilling the usual constraints $\alpha\le a\le\beta$ and $\displaystyle\int_D a(x)\,dx\le m$, which minimizes a cost function of the form $$\int_\Omega\int_Dj\big(x,\omega,u_a(x,\omega)\big)\,dx\,dP(\omega).$$ Some numerical examples are shown in the last section, which evidence the previous analytical results.
2011, 31(4): 1129-1150 doi: 10.3934/dcds.2011.31.1129 +[Abstract](479) +[PDF](1209.9KB)
Abstract:
We prove density estimates and elimination properties for minimizing triplets of functionals which are related to contour detection in image segmentation and depend on free discontinuities, free gradient discontinuities and second order derivatives. All the estimates concern optimal segmentation under Dirichlet boundary conditions and are uniform in the image domain up to the boundary.
2011, 31(4): 1151-1195 doi: 10.3934/dcds.2011.31.1151 +[Abstract](549) +[PDF](622.7KB)
Abstract:
We prove existence and uniqueness of entropy solutions of the Cauchy problem for the quasilinear parabolic equation $u_t$ $= div$ $a$$(u,Du) with initial condition u_0 \in BV(\mathbb{R}^N), u_0$$\geq 0$, where $a(z,\xi)$ = $\nabla_\xi f(z,\xi)$ and $f$ is a convex function of $\xi$ with linear growth as $\Vert \xi\Vert \to\infty$, satisfying other additional assumptions that cover the case of the so-called relativistic heat equation and other flux limited diffusion equations used in the theory of radiation hydrodynamics.
2011, 31(4): 1197-1218 doi: 10.3934/dcds.2011.31.1197 +[Abstract](398) +[PDF](578.4KB)
Abstract:
We study the possible regularization of collision solutions for one centre problems with a weak singularity. In the case of logarithmic singularities, we consider the method of regularization via smoothing of the potential. With this technique, we prove that the extended flow, where collision solutions are replaced with transmission trajectories, is continuous, though not differentiable, with respect to the initial data.
2011, 31(4): 1219-1231 doi: 10.3934/dcds.2011.31.1219 +[Abstract](526) +[PDF](391.0KB)
Abstract:
We present a recent existence result concerning the quasistatic evolution of cracks in hyperelastic brittle materials, in the framework of finite elasticity with non-interpenetration. In particular, here we consider the problem where no Dirichlet conditions are imposed, the boundary is traction-free, and the body is subject only to time-dependent volume forces. This allows us to present the main ideas of the proof in a simpler way, avoiding some of the technicalities needed in the general case, studied in [9].
2011, 31(4): 1233-1248 doi: 10.3934/dcds.2011.31.1233 +[Abstract](555) +[PDF](374.5KB)
Abstract:
We describe a direct variational approach to a class of semilinear elliptic equations with measure data. Using a typical variational argument, we show the existence of multiple solutions.
2011, 31(4): 1249-1272 doi: 10.3934/dcds.2011.31.1249 +[Abstract](517) +[PDF](481.3KB)
Abstract:
Following Almgren's construction of the center manifold in his Big regularity paper, we show the $C^{3,\alpha}$ regularity of area-minimizing currents in the neighborhood of points of density one without using the nonparametric theory. This study is intended as a first step towards the understanding of Almgren's construction in its full generality.
2011, 31(4): 1273-1292 doi: 10.3934/dcds.2011.31.1273 +[Abstract](439) +[PDF](389.4KB)
Abstract:
Usually, positively homogeneous functions are studied by means of exhaustive families of upper and lower approximations and their duals - upper and lower exhausters. Upper exhausters are used to find minimizers while lower exhausters are employed to find maximizers. In the paper, some properties of the so-called conversion operator (which converts an upper exhauster into a lower one, and vice versa) are discussed. The notions of cycle of exhausters, minimal cycle of exhausters and equivalent exhausters are introduced. A conjecture is formulated claiming that in the case of polyhedral exhausters only 1-cycle minimal exhausters exist.
2011, 31(4): 1293-1305 doi: 10.3934/dcds.2011.31.1293 +[Abstract](575) +[PDF](356.3KB)
Abstract:
Some problems of elasticity and shell theory are considered. The common feature of these problems is the presence of a small parameter $\varepsilon$. If $\varepsilon>0$ the corresponding equations are elliptic and the boundary conditions satisfy the Shapiro - Lopatinsky condition. When $\varepsilon=0$, this condition is violated and the problem can be non-solvable in the distribution spaces. The rather difficult passing to the limit is studied using the related Cauchy problem for elliptic equations. This approach allows to show that the most important is the transition zone where the frequencies $|\xi|\asymp \log (\varepsilon^{-1})$.
2011, 31(4): 1307-1323 doi: 10.3934/dcds.2011.31.1307 +[Abstract](539) +[PDF](412.3KB)
Abstract:
Let $\Omega$ be a bounded convex open set of $\mathbb{R}^n,$ $n\geq 2,$ $\partial \Omega$ of class $C^{2,1}.$ We consider the following Dirichlet problem $$\left\{\begin{array}{l} u\in H^2\cap H^1_0(\Omega,\mathbb{R}^N) \\ F(x,D^2 u(x))= f(x), \quad \text{a.e. in} \,\,\,\Omega, \end{array} \right.$$ where $f\in {\mathcal L}^{2,\lambda}(\Omega,\mathbb{R}^N),$ $n \leq$ $\lambda< n+2$, $F$ satisfies Campanato's Condition $A_x$ and is Hölder continuous in $\Omega$ with exponent $b.$
We show that there exist $\varepsilon, \overline{\varepsilon}\in (0,1),$ ($\varepsilon,\overline{\varepsilon}$ depend on $\gamma$ and $\delta$), such that for any $\zeta \in (0,\overline{\varepsilon}\, n) ,$ and $\mu \in( 0,\lambda],$ with $\mu< (2b+\zeta)\wedge [\epsilon\,(n+2)],$ we have $D^2 u \in {\mathcal L}^{2,\mu}(\Omega,\mathbb{R}^{n^2N}),$ where $\varepsilon$ and $\overline{\varepsilon}$ depend on the constants appearing in Condition $A_x.$
2011, 31(4): 1325-1346 doi: 10.3934/dcds.2011.31.1325 +[Abstract](490) +[PDF](412.4KB)
Abstract:
In this paper we study the properties of curves minimizing mechanical Lagrangian where the potential is Sobolev. Since a Sobolev function is only defined almost everywhere, no pointwise results can be obtained in this framework, and our point of view is shifted from single curves to measures in the space of paths. This study is motived by the goal of understanding the properties of variational solutions to the incompressible Euler equations.
2011, 31(4): 1347-1363 doi: 10.3934/dcds.2011.31.1347 +[Abstract](574) +[PDF](401.0KB)
Abstract:
Global Hölder regularity of the gradient in Morrey spaces is established for planar elliptic discontinuous equations, estimating in an explicit way the Hölder exponent in terms of the eigenvalues of the operator. The result is proved for Dirichlet or normal derivative problems and for nonlinear operators.
2011, 31(4): 1365-1381 doi: 10.3934/dcds.2011.31.1365 +[Abstract](568) +[PDF](464.7KB)
Abstract:
We establish pointwise decay bounds for radial, compact solutions of energy supercritical wave equations in odd dimensions. Applications are given.
2011, 31(4): 1383-1396 doi: 10.3934/dcds.2011.31.1383 +[Abstract](674) +[PDF](356.6KB)
Abstract:
In this paper we develop a new and efficient method for variational inequality with Lipschitz continuous strongly monotone operator. Our analysis is based on a new strongly convex merit function. We apply a variant of the developed scheme for solving quasivariational inequalities. As a result, we significantly improve the standard sufficient condition for existence and uniqueness of their solutions. Moreover, we get a new numerical scheme, whose rate of convergence is much higher than that of the straightforward gradient method.
2011, 31(4): 1397-1410 doi: 10.3934/dcds.2011.31.1397 +[Abstract](501) +[PDF](436.1KB)
Abstract:
We derive weak solvability and higher integrability of the spatial gradient of solutions to Cauchy--Dirichlet problem for divergence form quasilinear parabolic equations $$$$\left\{\begin{array}{l} u_t-\mathrm{div\,}\big(a^{ij}(x,t,u)D_ju+a^i(x,t,u)\big)=b(x,t,u,Du) &\quad \text{in}\ Q,\\ u=0 &\quad \text{on}\ \partial_p Q, \end{array} \right.$$$$ where $Q$ is a cylinder in $\mathbb{R}^n\times(0,T)$ with Reifenberg flat base $\Omega.$ The principal coefficients $a^{ij}(x,t,u)$ of the uniformly parabolic operator are supposed to have small $BMO$ norms with respect to $(x,t)$ while the nonlinear terms $a^i(x,t,u)$ and $b(x,t,u,Du)$ support controlled growth conditions.
2011, 31(4): 1411-1425 doi: 10.3934/dcds.2011.31.1411 +[Abstract](448) +[PDF](421.0KB)
Abstract:
The note concerns on some estimates in Morrey Spaces for the derivatives of local minimizers of variational integrals of the form $$\int_\Omega F (x,u,Du) dx$$ where the integrand has the following special form $$F(x,u,Du)\, =\, A(x,u, g^{\alpha\beta}(x) h_{ij}(u) \frac{\partial u^i}{\partial x^\alpha} \frac{\partial u^i }{\partial x^\beta}),$$ where $(g^{\alpha\beta})$ and $(h_{ij})$ symmetric positive definite matrices. We are not assuming the continuity of $A$ and $g$ with respect to $x$. We suppose that $A(\cdot, u,t)/(1+t)$ and $g(\cdot)$ are in the class $L^\infty\cap VMO$.
2011, 31(4): 1427-1451 doi: 10.3934/dcds.2011.31.1427 +[Abstract](693) +[PDF](496.4KB)
Abstract:
We are concerned with $\Gamma$-convergence of gradient flows, which is a notion meant to ensure that if a family of energy functionals depending of a parameter $\Gamma$-converges, then the solutions to the associated gradient flows converge as well. In this paper we present both a review of the abstract "theory" and of the applications it has had, and a generalization of the scheme to metric spaces which has not appeared elsewhere. We also mention open problems and perspectives.
2011, 31(4): 1453-1468 doi: 10.3934/dcds.2011.31.1453 +[Abstract](606) +[PDF](490.1KB)
Abstract:
In this paper, by providing simple counterexamples, several important results in bi-duality, triality and tri-duality, an optimization theory established and presented by D.Y. Gao in his book "Duality Principles in Nonconvex Systems. Theory, Methods and Applications," Kluwer Academic Publishers, Dordrecht, 2000, are proven to be false. Other results concerning this optimization theory from subsequent papers by D.Y. Gao and his collaborators are analyzed, false claims are exposed and when possible corrected, while the possibility or impossibility of obtaining correct various alternatives to the classical minimax relations are discussed.
2011, 31(4): 1469-1477 doi: 10.3934/dcds.2011.31.1469 +[Abstract](442) +[PDF](350.9KB)
Abstract:
In this paper, we investigate derivation in proper Jordan $CQ^{*}$-algebras associated with the following Pexiderized Jensen type functional equation $kf(\frac{x+y}{k}) = f_{0}(x)+ f_{1} (y).$ This is applied to investigate derivations and their Hyers--Ulam--Rassias stability in proper Jordan $CQ^{*}$-algebras.

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