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Discrete & Continuous Dynamical Systems - A

2011 , Volume 30 , Issue 2

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

Select all articles


Luis Caffarelli and  YanYan Li
2011, 30(2): i-ii doi: 10.3934/dcds.2011.30.2i +[Abstract](39) +[PDF](85.6KB)
"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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Attaching maps in the standard geodesics problem on $S^2$
Abbas Bahri
2011, 30(2): 379-426 doi: 10.3934/dcds.2011.30.379 +[Abstract](31) +[PDF](3490.5KB)
Unstable manifolds of critical points at infinity in the variational problems relating to periodic orbits of Reeb vector-fields in Contact Form Geometry are viewed in this paper as part of the attaching maps along which these variational problems attach themselves to natural generalizations that they have. The specific periodic orbit problem for the Reeb vector-field $\xi_0$ of the standard contact structure/form of $S^3$ is studied; the extended variational problem is the closed geodesics problem on $S^2$. The attaching maps are studied for low-dimensional (at most $4$) cells. Some circle and ''loop" actions on the loop space of $S^3$, that are lifts (via Hopf-fibration map) of the standard $S^1$-action on the free loop space of $S^2$, are also defined. ''Conjugacy" relations relating these actions are established.
An entropy based theory of the grain boundary character distribution
Katayun Barmak , Eva Eggeling , Maria Emelianenko , Yekaterina Epshteyn , David Kinderlehrer , Richard Sharp and  Shlomo Ta'asan
2011, 30(2): 427-454 doi: 10.3934/dcds.2011.30.427 +[Abstract](107) +[PDF](887.4KB)
Cellular networks are ubiquitous in nature. They exhibit behavior on many different length and time scales and are generally metastable. Most technologically useful materials are polycrystalline microstructures composed of a myriad of small monocrystalline grains separated by grain boundaries. The energetics and connectivity of the grain boundary network plays a crucial role in determining the properties of a material across a wide range of scales. A central problem in materials science is to develop technologies capable of producing an arrangement of grains—a texture—appropriate for a desired set of material properties. Here we discuss the role of energy in texture development, measured by a character distribution. We derive an entropy based theory based on mass transport and a Kantorovich-Rubinstein-Wasserstein metric to suggest that, to first approximation, this distribution behaves like the solution to a Fokker-Planck Equation.
Elliptic equations and systems with critical Trudinger-Moser nonlinearities
Djairo G. De Figueiredo , João Marcos do Ó and  Bernhard Ruf
2011, 30(2): 455-476 doi: 10.3934/dcds.2011.30.455 +[Abstract](58) +[PDF](486.4KB)
In this article we give first a survey on recent results on some Trudinger-Moser type inequalities, and their importance in the study of nonlinear elliptic equations with nonlinearities which have critical growth in the sense of Trudinger-Moser. Furthermore, recent results concerning systems of such equations will be discussed.
Interpolation by linear programming I
Charles Fefferman
2011, 30(2): 477-492 doi: 10.3934/dcds.2011.30.477 +[Abstract](32) +[PDF](399.7KB)
Given $m , n \geq 2$ and $\epsilon > 0$, we compute a function taking prescribed values at $N$ given points of $\mathbb{R}^n$, and having $C^m$ norm as small as possible up to a factor $1 + \epsilon$. Our computation reduces matters to a linear programming problem.
Fundamental solutions for a class of Isaacs integral operators
Patricio Felmer and  Alexander Quaas
2011, 30(2): 493-508 doi: 10.3934/dcds.2011.30.493 +[Abstract](35) +[PDF](383.9KB)
In this article we study the existence of fundamental solutions for a class of Isaacs integral operators and we apply them to prove Liouville type theorems. In proving these theorems we use the comparison principle for non-local operators.
Scale-invariant extinction time estimates for some singular diffusion equations
Yoshikazu Giga and  Robert V. Kohn
2011, 30(2): 509-535 doi: 10.3934/dcds.2011.30.509 +[Abstract](38) +[PDF](498.6KB)
We study three singular parabolic evolutions: the second-order total variation flow, the fourth-order total variation flow, and a fourth-order surface diffusion law. Each has the property that the solution becomes identically zero in finite time. We prove scale-invariant estimates for the extinction time, using a simple argument which combines an energy estimate with a suitable Sobolev-type inequality.
Nirenberg's contributions to complex analysis
Joseph J Kohn
2011, 30(2): 537-545 doi: 10.3934/dcds.2011.30.537 +[Abstract](43) +[PDF](357.2KB)
This article is concerned with the fundamental contributions of Louis Nirenberg to complex analysis and their impact on the theory of partial differential equations. We explain some of his main results and sketch the developments that they engendered.
Decay estimation for positive solutions of a $\gamma$-Laplace equation
Yutian Lei , Congming Li and  Chao Ma
2011, 30(2): 547-558 doi: 10.3934/dcds.2011.30.547 +[Abstract](36) +[PDF](390.1KB)
In this paper, we study the properties of the positive solutions of a $\gamma$-Laplace equation in $R^n$

-div$(|\nabla u|^{\gamma-2}\nabla u) =K u^p$,

Here $1<\gamma<2$, $n>\gamma$, $p=\frac{(\gamma-1)(n+\gamma)}{n-\gamma}$ and $K(x)$ is a smooth function bounded by two positive constants. First, the positive solution $u$ of the $\gamma$-Laplace equation above satisfies an integral equation involving a Wolff potential. Based on this, we estimate the decay rate of the positive solutions of the $\gamma$-Laplace equation at infinity. A new method is introduced to fully explore the integrability result established recently by Ma, Chen and Li on Wolff type integral equations to derive the decay estimate.

Regularity of optimal transport and cut locus: From nonsmooth analysis to geometry to smooth analysis
Cédric Villani
2011, 30(2): 559-571 doi: 10.3934/dcds.2011.30.559 +[Abstract](44) +[PDF](442.1KB)
In this survey paper I describe the convoluted links between the regularity theory of optimal transport and the geometry of cut locus.

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