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Discrete & Continuous Dynamical Systems - A

2010 , Volume 28 , Issue 2

A special issue
Dedicated to Louis Nirenberg on the Occasion of his 85th Birthday
Part I

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Preface
Luis A. Caffarelli and YanYan Li
2010, 28(2): i-ii doi: 10.3934/dcds.2010.28.2i +[Abstract](356) +[PDF](27.2KB)
Abstract:
"One of the wonders of mathematics is you go somewhere in the world and you meet other mathematicians, and it is like one big family. This large family is a wonderful joy."
   Louis Nirenberg, in an interview in the Notices of the AMS, April 2002.
   Louis Nirenberg was born in Hamilton, Ontario on February 28, 1925. He was attracted to physics as a high school student in Montreal while attending the Baron Byng School. He completed a major in Mathematics and Physics at McGill University. Having met Richard Courant, he went to graduate school at NYU and what would become the Courant Institute. There he completed his PhD degree under the direction of James Stoker. He was then invited to join the faculty and has been there ever since. He was one of the founding members of the Courant Institute of Mathematical Sciences and is now an Emeritus Professor.

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Optimal estimates for the gradient of harmonic functions in the multidimensional half-space
Gershon Kresin and Vladimir Maz’ya
2010, 28(2): 425-440 doi: 10.3934/dcds.2010.28.425 +[Abstract](310) +[PDF](214.4KB)
Abstract:
A representation of the sharp constant in a pointwise estimate of the gradient of a harmonic function in a multidimensional half-space is obtained under the assumption that function's boundary values belong to $L^p$. This representation is concretized for the cases $p=1, 2,$ and $\infty$.
Properties of translating solutions to mean curvature flow
Changfeng Gui, Huaiyu Jian and Hongjie Ju
2010, 28(2): 441-453 doi: 10.3934/dcds.2010.28.441 +[Abstract](411) +[PDF](204.6KB)
Abstract:
In this paper, we study the convexity, interior gradient estimate, Liouville type theorem and asymptotic behavior at infinity of translating solutions to mean curvature flow as well as the nonlinear flow by powers of the mean curvature.
Monotonicity methods for infinite dimensional sandwich systems
Martin Schechter
2010, 28(2): 455-468 doi: 10.3934/dcds.2010.28.455 +[Abstract](409) +[PDF](193.1KB)
Abstract:
We show how hypotheses for many problems can be significantly reduced if we employ the monotonicity method. We apply it to problems for the semilinear wave equation, where infinite dimensional methods are needed.
A global compactness result for the p-Laplacian involving critical nonlinearities
Carlo Mercuri and Michel Willem
2010, 28(2): 469-493 doi: 10.3934/dcds.2010.28.469 +[Abstract](455) +[PDF](267.8KB)
Abstract:
We prove a representation theorem for Palais-Smale sequences involving the p-Laplacian and critical nonlinearities. Applications are given to a critical problem.
On some strong ratio limit theorems for heat kernels
Martin Fraas, David Krejčiřík and Yehuda Pinchover
2010, 28(2): 495-509 doi: 10.3934/dcds.2010.28.495 +[Abstract](440) +[PDF](219.7KB)
Abstract:
We study strong ratio limit properties of the quotients of the heat kernels of subcritical and critical operators which are defined on a noncompact Riemannian manifold.
The validity of the Euler-Lagrange equation
Giovanni Bonfanti and Arrigo Cellina
2010, 28(2): 511-517 doi: 10.3934/dcds.2010.28.511 +[Abstract](400) +[PDF](126.9KB)
Abstract:
We prove the validity of the Euler-Lagrange equation for a solution $u$ to the problem of minimizing $\int_{\Omega}L(x,u(x),\nabla u(x))dx$, where $L$ is a Carathéodory function, convex in its last variable, without assuming differentiability with respect to this variable.
A variational problem in the mechanics of complex materials
Mariano Giaquinta, Paolo Maria Mariano and Giuseppe Modica
2010, 28(2): 519-537 doi: 10.3934/dcds.2010.28.519 +[Abstract](504) +[PDF](270.7KB)
Abstract:
We analyze the possible nucleation of cracked surfaces in materials in which changes in the material texture have a prominent influence on the macroscopic mechanical behavior. The geometry of crack patterns is described by means of stratified families of curvature varifolds with boundary. Possible non-local actions of the microstructures are accounted for. We prove existence of ground states of the energy in terms of deformation, descriptors of the microstructure and varifolds.
Glaeser's type gradient estimates for non-negative solutions of fully nonlinear elliptic equations
Italo Capuzzo Dolcetta and Antonio Vitolo
2010, 28(2): 539-557 doi: 10.3934/dcds.2010.28.539 +[Abstract](395) +[PDF](275.4KB)
Abstract:
In this paper we discuss some extensions to a fully nonlinear setting of results by Y.Y. Li and L. Nirenberg [25] about gradient estimates for non-negative solutions of linear elliptic equations. Our approach relies heavily on methods developed by L. Caffarelli in [3] and [4].
Partial regularity of Brenier solutions of the Monge-Ampère equation
Alessio Figalli and Young-Heon Kim
2010, 28(2): 559-565 doi: 10.3934/dcds.2010.28.559 +[Abstract](500) +[PDF](148.8KB)
Abstract:
Given $\Omega,\Lambda \subset \R^n$ two bounded open sets, and $f$ and $g$ two probability densities concentrated on $\Omega$ and $\Lambda$ respectively, we investigate the regularity of the optimal map $\nabla \varphi$ (the optimality referring to the Euclidean quadratic cost) sending $f$ onto $g$. We show that if $f$ and $g$ are both bounded away from zero and infinity, we can find two open sets $\Omega'\subset \Omega$ and $\Lambda'\subset \Lambda$ such that $f$ and $g$ are concentrated on $\Omega'$ and $\Lambda'$ respectively, and $\nabla\varphi:\Omega' \to \Lambda'$ is a (bi-Hölder) homeomorphism. This generalizes the $2$-dimensional partial regularity result of [8].
Charged cosmological dust solutions of the coupled Einstein and Maxwell equations
Joel Spruck and Yisong Yang
2010, 28(2): 567-589 doi: 10.3934/dcds.2010.28.567 +[Abstract](344) +[PDF](299.9KB)
Abstract:
It is well known through the work of Majumdar, Papapetrou, Hartle, and Hawking that the coupled Einstein and Maxwell equations admit a static multiple blackhole solution representing a balanced equilibrium state of finitely many point charges. This is a result of the exact cancellation of gravitational attraction and electric repulsion under an explicit condition on the mass and charge ratio. The resulting system of particles, known as an extremely charged dust, gives rise to examples of spacetimes with naked singularities. In this paper, we consider the continuous limit of the Majumdar-Papapetrou-Hartle-Hawking solution modeling a space occupied by an extended distribution of extremely charged dust. We show that for a given smooth distribution of matter of finite ADM mass there is a continuous family of smooth solutions realizing asymptotically flat space metrics.
Sets with finite perimeter in Wiener spaces, perimeter measure and boundary rectifiability
Luigi Ambrosio, Michele Miranda jr. and Diego Pallara
2010, 28(2): 591-606 doi: 10.3934/dcds.2010.28.591 +[Abstract](391) +[PDF](229.4KB)
Abstract:
We discuss some recent developments of the theory of $BV$ functions and sets of finite perimeter in infinite-dimensional Gaussian spaces. In this context the concepts of Hausdorff measure, approximate continuity, rectifiability have to be properly understood. After recalling the known facts, we prove a Sobolev-rectifiability result and we list some open problems.
On the local solvability of the Nirenberg problem on $\mathbb S^2$
Zheng-Chao Han and YanYan Li
2010, 28(2): 607-615 doi: 10.3934/dcds.2010.28.607 +[Abstract](406) +[PDF](170.4KB)
Abstract:
We present some results on the local solvability of the Nirenberg problem on $\mathbb S^2$. More precisely, an $L^2(\mathbb S^2)$ function near $1$ is the Gauss curvature of an $H^2(\mathbb S^2)$ metric on the round sphere $\mathbb S^2$, pointwise conformal to the standard round metric on $\mathbb S^2$, provided its $L^2(\mathbb S^2)$ projection into the the space of spherical harmonics of degree $2$ satisfy a matrix invertibility condition, and the ratio of the $L^2(\mathbb S^2)$ norms of its $L^2(\mathbb S^2)$ projections into the the space of spherical harmonics of degree $1$ vs the space of spherical harmonics of degrees other than $1$ is sufficiently small.
Nodal minimal partitions in dimension $3$
Bernard Helffer, Thomas Hoffmann-Ostenhof and Susanna Terracini
2010, 28(2): 617-635 doi: 10.3934/dcds.2010.28.617 +[Abstract](381) +[PDF](270.6KB)
Abstract:
In continuation of [20], we analyze the properties of spectral minimal $k$-partitions of an open set $\Omega$ in $\mathbb R^3$ which are nodal, i.e. produced by the nodal domains of an eigenfunction of the Dirichlet Laplacian in $\Omega$. We show that such a partition is necessarily a nodal partition associated with a $k$-th eigenfunction. Hence we have in this case equality in Courant's nodal theorem.
Radial solutions for Neumann problems with $\phi$-Laplacians and pendulum-like nonlinearities
Cristian Bereanu, Petru Jebelean and Jean Mawhin
2010, 28(2): 637-648 doi: 10.3934/dcds.2010.28.637 +[Abstract](369) +[PDF](183.6KB)
Abstract:
In this paper we study the existence and multiplicity of radial solutions for Neumann problems in a ball and in an annular domain, associated to pendulum-like perturbations of mean curvature operators in Euclidean and Minkowski spaces and of the $p$-Laplacian operator. Our approach relies on the Leray-Schauder degree and the upper and lower solutions method.
Existence of solutions for a semilinear wave equation with non-monotone nonlinearity
Alfonso Castro and Benjamin Preskill
2010, 28(2): 649-658 doi: 10.3934/dcds.2010.28.649 +[Abstract](449) +[PDF](165.4KB)
Abstract:
For double-periodic and Dirichlet-periodic boundary conditions, we prove the existence of solutions to a forced semilinear wave equation with asymptotically linear nonlinearity, no resonance, and non-monotone nonlinearity when the forcing term is not flat on characteristics. The solutions are in $L^{\infty}$ when the forcing term is in $L^{\infty}$ and continous when the forcing term is continuous. This is in contrast with the results in [4], where the non-enxistence of continuous solutions is established even when forcing term is of class $C^{\infty}$ but is flat on a characteristic.
A Liouville problem for the Sigma-2 equation
Sun-Yung Alice Chang and Yu Yuan
2010, 28(2): 659-664 doi: 10.3934/dcds.2010.28.659 +[Abstract](347) +[PDF](102.6KB)
Abstract:
We show that any global convex solution to the Sigma-2 equation must be quadratic.
Sc-smoothness, retractions and new models for smooth spaces
Helmut Hofer, Kris Wysocki and Eduard Zehnder
2010, 28(2): 665-788 doi: 10.3934/dcds.2010.28.665 +[Abstract](413) +[PDF](1015.3KB)
Abstract:
We present the concept of sc-smoothness for Banach spaces, which leads to new models of spaces having locally varying dimensions called M-polyfolds. We present detailed proofs of the technical results needed for the applications, in particular, to the Symplectic Field Theory. We also outline a very general Fredholm theory for bundles over M-polyfolds. The concepts are illustrated by holomorphic mappings between conformal cylinders which break apart as the modulus tends to infinity.
A structural condition for microscopic convexity principle
Baojun Bian and Pengfei Guan
2010, 28(2): 789-807 doi: 10.3934/dcds.2010.28.789 +[Abstract](364) +[PDF](238.2KB)
Abstract:
The arguments in paper [2] have been refined to prove a microscopic convexity principle for fully nonlinear elliptic equation under a more natural structure condition. We also consider the corresponding result for the partial convexity case.
On a new index theory and non semi-trivial solutions for elliptic systems
Kung-Ching Chang, Zhi-Qiang Wang and Tan Zhang
2010, 28(2): 809-826 doi: 10.3934/dcds.2010.28.809 +[Abstract](360) +[PDF](223.5KB)
Abstract:
Two indices, which are similar to the Krasnoselski's genus on the sphere, are defined on the product of spheres. They are applied to investigate the multiple non semi-trivial solutions for elliptic systems. Both constraint and unconstraint problems are studied.
On some Schrödinger equations with non regular potential at infinity
Giovanna Cerami and Riccardo Molle
2010, 28(2): 827-844 doi: 10.3934/dcds.2010.28.827 +[Abstract](435) +[PDF](241.6KB)
Abstract:
In this paper we study the existence of solutions $u\in H^1(\R^N)$ for the problem $-\Delta u+a(x)u=|u|^{p-2}u$, where $N\ge 2$ and $p$ is superlinear and subcritical. The potential $a(x)\in L^\infty(\R^N)$ is such that $a(x)\ge c>0$ but is not assumed to have a limit at infinity. Considering different kinds of assumptions on the geometry of $a(x)$ we obtain two theorems stating the existence of positive solutions. Furthermore, we prove that there are no nontrivial solutions, when a direction exists along which the potential is increasing.
A Neumann eigenvalue problem for fully nonlinear operators
Isabeau Birindelli and Stefania Patrizi
2010, 28(2): 845-863 doi: 10.3934/dcds.2010.28.845 +[Abstract](380) +[PDF](252.0KB)
Abstract:
We prove the existence of the principal eigenvalues for the Pucci operators in bounded domains with boundary condition $\frac{\partial u}{\partial\vec n}=\alpha u$ corresponding respectively to positive and negative eigenfunctions and study their asymptotic behavior when $\alpha$ goes to $+\infty$.

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