# American Institute of Mathematical Sciences

ISSN:
1078-0947

eISSN:
1553-5231

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## Discrete & Continuous Dynamical Systems - A

2010 , Volume 28 , Issue 3

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

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2010, 28(3): i-ii doi: 10.3934/dcds.2010.28.3i +[Abstract](474) +[PDF](27.4KB)
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.

2010, 28(3): 865-873 doi: 10.3934/dcds.2010.28.865 +[Abstract](627) +[PDF](141.5KB)
Abstract:
We prove that global Lipschitz solutions to the linearized Monge-Ampere equation

$L_$φ$u$:$=\sum$φij$u_{ij}=0$

must be linear in $2D$. The function φ is assumed to have the Monge-Ampere measure $\det D^2$φ bounded away from $0$ and $\infty$.

2010, 28(3): 875-897 doi: 10.3934/dcds.2010.28.875 +[Abstract](560) +[PDF](282.2KB)
Abstract:
This paper is devoted to the study of solitons whose existence is related to the ratio energy/charge. These solitons are called hylomorphic. In the first part of the paper we prove an abstract theorem on the existence of hylomorphic solitons which can be applied to the main situations considered in literature. In the second part, we apply this theorem to the nonlinear Schrödinger and Klein Gordon equations defined on a lattice.
2010, 28(3): 899-913 doi: 10.3934/dcds.2010.28.899 +[Abstract](527) +[PDF](197.3KB)
Abstract:
In this paper we study partial and anisotropic Schauder estimates for linear and nonlinear elliptic equations. We prove that if the inhomogeneous term $f$ is Hölder continuous in the $x_n$-direction, then the mixed derivatives uxxn are Hölder continuous; if $f$ satisfies an anisotropic Hölder continuity condition, then the second derivatives $D^2 u$ satisfy related anisotropic Hölder continuity estimates.
2010, 28(3): 915-929 doi: 10.3934/dcds.2010.28.915 +[Abstract](511) +[PDF](223.4KB)
Abstract:
In this paper, we study the local structure and the smoothness of singularities of free boundaries in an optimal partition problem for the Dirichlet eigenvalues. We prove that there is a unique homogeneous blow up(tangent map) at each singular point in the interior of the free boundary. As a consequence we obtain the rectifiability as well as local structures of singularities.
2010, 28(3): 931-973 doi: 10.3934/dcds.2010.28.931 +[Abstract](467) +[PDF](561.0KB)
Abstract:
We present some recent results on mean field equations of Liouville type over a closed surface, in presence of Dirac distributions supported at the so called "vortex points". We discuss possible existence and non-existence results as well as uniqueness and multiplicity issues according to the topological and geometrical properties of the surface.
2010, 28(3): 975-1006 doi: 10.3934/dcds.2010.28.975 +[Abstract](589) +[PDF](396.8KB)
Abstract:
Let (M ,ğ) be an $N$-dimensional smooth (compact or noncompact) Riemannian manifold. We introduce the elliptic Jacobi-Toda system on (M ,ğ). We review various recent results on its role in the construction of solutions with multiple interfaces of the Allen-Cahn equation on compact manifolds and entire space, as well as multiple-front traveling waves for its parabolic counterpart.
2010, 28(3): 1007-1031 doi: 10.3934/dcds.2010.28.1007 +[Abstract](539) +[PDF](291.5KB)
Abstract:
We follow up our work [4] concerning the formation of trapped surfaces. We provide a considerable extension of our result there on pre-scared surfaces to allow for the formation of a surface with multiple pre-scared angular regions which, together, can cover an arbitrarily large portion of the surface. In a forthcoming paper we plan to show that once a significant part of the surface is pre-scared, it can be additionally deformed to produce a bona-fide trapped surface.
2010, 28(3): 1033-1050 doi: 10.3934/dcds.2010.28.1033 +[Abstract](558) +[PDF](239.3KB)
Abstract:
We examine the regularity of the extremal solution of the nonlinear eigenvalue problem $\Delta^2 u = \lambda f(u)$ on a general bounded domain $\Omega$ in $\R^N$, with the Navier boundary condition $u=\Delta u =0$ on δΩ. We establish energy estimates which show that for any non-decreasing convex and superlinear nonlinearity $f$ with $f(0)=1$, the extremal solution u * is smooth provided $N\leq 5$. If in addition $\lim$i$nf_{t \to +\infty}\frac{f (t)f'' (t)}{(f')^2(t)}>0$, then u * is regular for $N\leq 7$, while if $\gamma$:$= \lim$s$up_{t \to +\infty}\frac{f (t)f'' (t)}{(f')^2(t)}<+\infty$, then the same holds for $N < \frac{8}{\gamma}$. It follows that u * is smooth if $f(t) = e^t$ and $N \le 8$, or if $f(t) = (1+t)^p$ and $N< \frac{8p}{p-1}$. We also show that if $f(t) = (1-t)^{-p}$, $p>1$ and $p\ne 3$, then u * is smooth for $N \leq \frac{8p}{p+1}$. While these results are major improvements on what is known for general domains, they still fall short of the expected optimal results as recently established on radial domains, e.g., u * is smooth for $N \le 12$ when $f(t) = e^t$ [11], and for $N \le 8$ when $f(t) = (1-t)^{-2}$ [9] (see also [22]).
2010, 28(3): 1051-1067 doi: 10.3934/dcds.2010.28.1051 +[Abstract](641) +[PDF](222.8KB)
Abstract:
Limiting profiles of solutions to a 2$\times$2 Lotka-Volterra competition-diffusion-advection system, when the strength of the advection tends to infinity, are determined. The two species, competing in a heterogeneous environment, are identical except for their dispersal strategies: One is just random diffusion while the other is "smarter" - a combination of random diffusion and a directed movement up the environmental gradient. With important progress made, it has been conjectured in [2] and [3] that for large advection the "smarter" species will concentrate near a selected subset of positive local maximum points of the environment function. In this paper, we establish this conjecture in one space dimension, with the peaks located and the limiting profiles determined, under mild hypotheses on the environment function.
2010, 28(3): 1069-1081 doi: 10.3934/dcds.2010.28.1069 +[Abstract](630) +[PDF](180.4KB)
Abstract:
We prove that the solution to a parabolic integro-differential equation with a gradient dependence that satisfies a critical power growth becomes immediately Hölder continuous. We also obtain some results in the supercritical case.
2010, 28(3): 1083-1099 doi: 10.3934/dcds.2010.28.1083 +[Abstract](606) +[PDF](239.5KB)
Abstract:
We consider the following semilinear elliptic equation on a strip:

$\Delta u-u + u^p=0 \ \mbox{in} \ \R^{N-1} \times (0, L),$
$u>0, \frac{\partial u}{\partial \nu}=0 \ \mbox{on} \ \partial (\R^{N-1} \times (0, L))$

where $1< p\leq \frac{N+2}{N-2}$. When $1 < p <\frac{N+2}{N-2}$, it is shown that there exists a unique L * >0 such that for L $\leq$L * , the least energy solution is trivial, i.e., doesn't depend on $x_N$, and for L >L * , the least energy solution is nontrivial. When $N \geq 4, p=\frac{N+2}{N-2}$, it is shown that there are two numbers L * < L ** such that the least energy solution is trivial when L $\leq$L *, the least energy solution is nontrivial when L $\in$(L *,L **], and the least energy solution does not exist when L >L **. A connection with Delaunay surfaces in CMC theory is also made.

2010, 28(3): 1101-1119 doi: 10.3934/dcds.2010.28.1101 +[Abstract](613) +[PDF](225.8KB)
Abstract:
In 1928, motivated by conversations with Keynes, Ramsey formulated an infinite-horizon problem in the calculus of variations. This problem is now classical in economic theory, and its solution lies at the heart of our understanding of economic growth. On the other hand, from the mathematical point of view, it was never solved in a satisfactory manner: In this paper, we give what we believe is the first complete mathematical treatment of the problem, and we show that its solution relies on solving an implicit differential equation. Such equations were first studied by Thom, and we use the geometric method he advocated. We then extend the Ramsey problem to non-constant discount rates, along the lines of Ekeland and Lazrak. In that case, there is time-inconsistency, meaning that optimal growth no longer is a relevant concept for economics, and has to be replaced with equlibrium growth. We briefly define what we mean by equilibrium growth, and proceed to prove that such a path actually exists, The problem, once again, reduces to solving an implicit differential equation, but this time the dimension is higher, and the analysis is more complicated: geometry is not enough, and we have to appeal to the central manifold theorem.
2010, 28(3): 1121-1135 doi: 10.3934/dcds.2010.28.1121 +[Abstract](666) +[PDF](216.8KB)
Abstract:
In this paper, we prove interior second derivative estimates of Pogorelov type for a general form of Monge-Ampère equation which includes the optimal transportation equation. The estimate extends that in a previous work with Xu-Jia Wang and assumes only that the matrix function in the equation is regular with respect to the gradient variables, that is it satisfies a weak form of the condition introduced previously by Ma,Trudinger and Wang for regularity of optimal transport mappings. We also indicate briefly an application to optimal transportation.
2010, 28(3): 1137-1150 doi: 10.3934/dcds.2010.28.1137 +[Abstract](529) +[PDF](219.9KB)
Abstract:
This is an expository paper dedicated to professor L. Nirenberg for his 85th birthday. First I will discuss my joint works with Z. Zhang and J. Song on the singularity formation of Kähler-Ricci flow. Secondly, I will show a fully nonlinear equation, scalar V-soliton equation (cf. Section 4, (14)), and some basic results about it. This equation was introduced by G. La Nave and myself in studying the singularity formation of Kähler-Ricci flow. I will also show how this new equation can be applied to studying the singularity formation at finite time.
2010, 28(3): 1151-1164 doi: 10.3934/dcds.2010.28.1151 +[Abstract](585) +[PDF](183.5KB)
Abstract:
We give a positive lower bound for the principal curvature of the strict convex level sets of harmonic functions in terms of the principal curvature of the domain boundary and the norm of the boundary gradient. We also extend this result to a class of semi-linear elliptic partial differential equations under certain structure condition.
2010, 28(3): 1165-1178 doi: 10.3934/dcds.2010.28.1165 +[Abstract](510) +[PDF](214.5KB)
Abstract:
A beautiful and influential subject in the study of the question of smoothness of solutions for the Navier - Stokes equations in three dimensions is the theory of partial regularity. A major paper on this topic is Caffarelli, Kohn & Nirenberg [5](1982) which gives an upper bound on the size of the singular set $S(u)$ of a suitable weak solution $u$. In the present paper we describe a complementary lower bound. More precisely, we study the situation in which a weak solution fails to be continuous in the strong $L^2$ topology at some singular time $t=T$. We identify a closed set in space on which the $L^2$ norm concentrates at this time $T$, and we study microlocal properties of the Fourier transform of the solution in the cotangent bundle T * (R 3) above this set. Our main result is that $L^2$ concentration can only occur on subsets of T * (R 3) which are sufficiently large. An element of the proof is a new global estimate on weak solutions of the Navier - Stokes equations which have sufficiently smooth initial data.
2010, 28(3): 1179-1206 doi: 10.3934/dcds.2010.28.1179 +[Abstract](565) +[PDF](333.2KB)
Abstract:
We establish sharp energy estimates for some solutions, such as global minimizers, monotone solutions and saddle-shaped solutions, of the fractional nonlinear equation $(-\Delta)$1/2 $u=f(u)$ in R n. Our energy estimates hold for every nonlinearity $f$ and are sharp since they are optimal for one-dimensional solutions, that is, for solutions depending only on one Euclidian variable.
As a consequence, in dimension $n=3$, we deduce the one-dimensional symmetry of every global minimizer and of every monotone solution. This result is the analog of a conjecture of De Giorgi on one-dimensional symmetry for the classical equation $-\Delta u=f(u)$ in R n.
2010, 28(3): 1207-1235 doi: 10.3934/dcds.2010.28.1207 +[Abstract](484) +[PDF](333.2KB)
Abstract:
We imbed an array of thin highly conductive fibers in a surrounding two-dimensional medium with small viscosity. The resulting composite medium is described by a second order elliptic operator in divergence form with discontinuous singular coefficients on an open domain of the plane. We study the asymptotic spectral behavior of the operator when, simultaneously, the viscosity vanishes and the fibers develop fractal geometry. We prove that the spectral measure of the operator converges to the spectral measure of a self-adjoint operator associated with the lower-dimensional fractal limit of the thin fibers. The limit fiber is a compact set that disconnects the initial domain into infinitely many non-empty open components. Our approach is of variational nature and relies on Hilbert space convergence of quadratic energy forms.
2010, 28(3): 1237-1272 doi: 10.3934/dcds.2010.28.1237 +[Abstract](663) +[PDF](396.9KB)
Abstract:
In this and the subsequent paper, we are interested in the following nonlinear equation:

$\Delta_g v+\rho(\frac{h^* e^v}{\int_M h^* vd\mu(x)}-1)= 4\pi\sum_{j=1}^N\alpha_j(\delta_{q_i}-1)\quad\text{in }M,$(0.1)

where $(M,g)$ is a Riemann surface with its area $|M|=1$; or

$\Delta v+\rho\frac{h^*e^v}{\int_\Omega h^* e^vdx}=4\pi\sum_{j=1}^N\alpha_j \delta_{q_j}\quad\text{in }\Omega,$ (0.2)

where $\Omega$ is a bounded smooth domain in $R^2$. Here, $\rho, \alpha_j$ are positive constants, $\delta_q$ is the Dirac measure at $q$, and both $h^*$'s are positive smooth functions. In this paper, we prove a sharp estimate for a sequence of blowing up solutions $u_k$ to (0.1) or (0.2) with $\rho_k\rightarrow\rho*. Among other things, we show that for equation (0.1),$\rho_k-\rho_*=\sum_{j=1}^\tau d_j( \Delta \log h^*(p_j)+\rho_*-N^*-2K(p_j)+o(1) )e^{-\frac{\lambda_k}{1+\alpha_j}}, $(0.3) and for equation (0.2),$ \rho_k-\rho_*=\sum_{j=1}^\tau d_j(\Delta \log h^*(p_j)+o(1))e^{-\frac{\lambda_k}{1+\alpha_j}},$(0.4) where$\lambda_k\rightarrow+\infty$and$d_j$is a constant depending on$p_j$, a blow up point of$u_k$. See section 1 for more precise description. These estimates play an important role when the degree counting formulas are derived. The subsequent paper [19] will complete the proof of computing the degree counting formula. 2010, 28(3): 1273-1290 doi: 10.3934/dcds.2010.28.1273 +[Abstract](518) +[PDF](230.2KB) Abstract: We study the local behavior of a solution to the Stokes system with singular coefficients in$R^n$with$n=2,3$. One of our main results is a bound on the vanishing order of a nontrivial solution$u$satisfying the Stokes system, which is a quantitative version of the strong unique continuation property for$u$. Different from the previous known results, our strong unique continuation result only involves the velocity field$u$. Our proof relies on some delicate Carleman-type estimates. We first use these estimates to derive crucial optimal three-ball inequalities for$u$. Taking advantage of the optimality, we then derive an upper bound on the vanishing order of any nontrivial solution$u$to the Stokes system from those three-ball inequalities. As an application, we derive a minimal decaying rate at infinity of any nontrivial$u$satisfying the Stokes equation under some a priori assumptions. 2010, 28(3): 1291-1298 doi: 10.3934/dcds.2010.28.1291 +[Abstract](616) +[PDF](131.4KB) Abstract: We prove two mixed versions of the Discrete Nodal Theorem of Davies et. al. [3] for bounded degree graphs, and for three-connected graphs of fixed genus$g$. Using this we can show that for a three-connected graph satisfying a certain volume-growth condition, the multiplicity of the$n$th Laplacian eigenvalue is at most$2[ 6(n-1) + 15(2g-2)]^2\$. Our results hold for any Schrödinger operator, not just the Laplacian.

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