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1078-0947
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Discrete & Continuous Dynamical Systems
June 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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2010, 28(2): i-ii
doi: 10.3934/dcds.2010.28.2i
+[Abstract](2589)
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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.
For more information please click the “Full Text” above.
"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.
For more information please click the “Full Text” above.
2010, 28(2): 425-440
doi: 10.3934/dcds.2010.28.425
+[Abstract](2209)
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Abstract:
A representation of the sharp constant in a pointwise estimate of the gradientof a harmonic function in a multidimensional half-space is obtained under the assumption thatfunction's boundary values belong to $L^p$. This representation is concretized for thecases $p=1, 2,$ and $\infty$.
A representation of the sharp constant in a pointwise estimate of the gradientof a harmonic function in a multidimensional half-space is obtained under the assumption thatfunction's boundary values belong to $L^p$. This representation is concretized for thecases $p=1, 2,$ and $\infty$.
2010, 28(2): 441-453
doi: 10.3934/dcds.2010.28.441
+[Abstract](2864)
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Abstract:
In this paper, we study the convexity, interior gradient estimate,Liouville type theorem and asymptotic behavior at infinity oftranslating solutions to mean curvature flow as well as thenonlinear flow by powers of the mean curvature.
In this paper, we study the convexity, interior gradient estimate,Liouville type theorem and asymptotic behavior at infinity oftranslating solutions to mean curvature flow as well as thenonlinear flow by powers of the mean curvature.
2010, 28(2): 455-468
doi: 10.3934/dcds.2010.28.455
+[Abstract](2395)
+[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.
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.
2010, 28(2): 469-493
doi: 10.3934/dcds.2010.28.469
+[Abstract](3920)
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Abstract:
We prove a representation theorem for Palais-Smale sequences involving the p-Laplacian and critical nonlinearities. Applications are given to a critical problem.
We prove a representation theorem for Palais-Smale sequences involving the p-Laplacian and critical nonlinearities. Applications are given to a critical problem.
2010, 28(2): 495-509
doi: 10.3934/dcds.2010.28.495
+[Abstract](2791)
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Abstract:
We study strong ratio limit properties of the quotients of the heat kernelsof subcritical and critical operators which are defined on a noncompact Riemannian manifold.
We study strong ratio limit properties of the quotients of the heat kernelsof subcritical and critical operators which are defined on a noncompact Riemannian manifold.
2010, 28(2): 511-517
doi: 10.3934/dcds.2010.28.511
+[Abstract](2793)
+[PDF](126.9KB)
Abstract:
We prove the validity of the Euler-Lagrange equationfor 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.
We prove the validity of the Euler-Lagrange equationfor 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.
2010, 28(2): 519-537
doi: 10.3934/dcds.2010.28.519
+[Abstract](2929)
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Abstract:
We analyze the possible nucleation of cracked surfaces in materials in whichchanges in the material texture have a prominent influence on themacroscopic mechanical behavior. The geometry of crackpatterns is described by means of stratified families of curvature varifoldswith boundary. Possible non-local actions of the microstructures areaccounted for. We prove existence of ground states of the energy in terms ofdeformation, descriptors of the microstructure and varifolds.
We analyze the possible nucleation of cracked surfaces in materials in whichchanges in the material texture have a prominent influence on themacroscopic mechanical behavior. The geometry of crackpatterns is described by means of stratified families of curvature varifoldswith boundary. Possible non-local actions of the microstructures areaccounted for. We prove existence of ground states of the energy in terms ofdeformation, descriptors of the microstructure and varifolds.
2010, 28(2): 539-557
doi: 10.3934/dcds.2010.28.539
+[Abstract](2936)
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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].
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].
2010, 28(2): 559-565
doi: 10.3934/dcds.2010.28.559
+[Abstract](3451)
+[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 findtwo 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 regularityresult of [8].
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 findtwo 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 regularityresult of [8].
2010, 28(2): 567-589
doi: 10.3934/dcds.2010.28.567
+[Abstract](2331)
+[PDF](299.9KB)
Abstract:
It is well known through the work of Majumdar, Papapetrou, Hartle, and Hawking that the coupledEinstein 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.
It is well known through the work of Majumdar, Papapetrou, Hartle, and Hawking that the coupledEinstein 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.
2010, 28(2): 591-606
doi: 10.3934/dcds.2010.28.591
+[Abstract](2461)
+[PDF](229.4KB)
Abstract:
We discuss some recent developments of the theory of $BV$ functionsand sets of finite perimeter in infinite-dimensional Gaussianspaces. In this context the concepts of Hausdorff measure,approximate continuity, rectifiability have to be properlyunderstood. After recalling the known facts, we prove aSobolev-rectifiability result and we list some open problems.
We discuss some recent developments of the theory of $BV$ functionsand sets of finite perimeter in infinite-dimensional Gaussianspaces. In this context the concepts of Hausdorff measure,approximate continuity, rectifiability have to be properlyunderstood. After recalling the known facts, we prove aSobolev-rectifiability result and we list some open problems.
2010, 28(2): 607-615
doi: 10.3934/dcds.2010.28.607
+[Abstract](2594)
+[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 standardround metric on $\mathbb S^2$, provided its $L^2(\mathbb S^2)$ projection into thethe space of spherical harmonics of degree $2$ satisfy a matrix invertibility condition,and the ratio of the $L^2(\mathbb S^2)$ norms ofits $L^2(\mathbb S^2)$ 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.
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 standardround metric on $\mathbb S^2$, provided its $L^2(\mathbb S^2)$ projection into thethe space of spherical harmonics of degree $2$ satisfy a matrix invertibility condition,and the ratio of the $L^2(\mathbb S^2)$ norms ofits $L^2(\mathbb S^2)$ 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.
2010, 28(2): 617-635
doi: 10.3934/dcds.2010.28.617
+[Abstract](2296)
+[PDF](270.6KB)
Abstract:
In continuation of [20], we analyze theproperties 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.
In continuation of [20], we analyze theproperties 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.
2010, 28(2): 637-648
doi: 10.3934/dcds.2010.28.637
+[Abstract](2533)
+[PDF](183.6KB)
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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.
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.
2010, 28(2): 649-658
doi: 10.3934/dcds.2010.28.649
+[Abstract](2742)
+[PDF](165.4KB)
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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 isflat on a characteristic.
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 isflat on a characteristic.
2010, 28(2): 659-664
doi: 10.3934/dcds.2010.28.659
+[Abstract](2330)
+[PDF](102.6KB)
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We show that any global convex solution to the Sigma-2 equation must be quadratic.
We show that any global convex solution to the Sigma-2 equation must be quadratic.
2010, 28(2): 665-788
doi: 10.3934/dcds.2010.28.665
+[Abstract](2538)
+[PDF](1015.3KB)
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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.
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.
2010, 28(2): 789-807
doi: 10.3934/dcds.2010.28.789
+[Abstract](2440)
+[PDF](238.2KB)
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The arguments in paper [2] have been refined to prove amicroscopic convexity principle for fully nonlinear ellipticequation under a more natural structure condition. We also consider the correspondingresult for the partial convexity case.
The arguments in paper [2] have been refined to prove amicroscopic convexity principle for fully nonlinear ellipticequation under a more natural structure condition. We also consider the correspondingresult for the partial convexity case.
2010, 28(2): 809-826
doi: 10.3934/dcds.2010.28.809
+[Abstract](2977)
+[PDF](223.5KB)
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Two indices, which are similar to the Krasnoselski's genus on thesphere, are defined on the product of spheres. They are applied toinvestigate the multiple non semi-trivial solutions for ellipticsystems. Both constraint and unconstraint problems are studied.
Two indices, which are similar to the Krasnoselski's genus on thesphere, are defined on the product of spheres. They are applied toinvestigate the multiple non semi-trivial solutions for ellipticsystems. Both constraint and unconstraint problems are studied.
2010, 28(2): 827-844
doi: 10.3934/dcds.2010.28.827
+[Abstract](2609)
+[PDF](241.6KB)
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In this paper we study the existence of solutions $u\in H^1(\R^N)$ forthe problem $-\Delta u+a(x)u=|u|^{p-2}u$, where $N\ge 2$ and $p$ issuperlinear and subcritical.The potential $a(x)\in L^\infty(\R^N)$ is such that $a(x)\ge c>0$ butis 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 adirection exists along which the potential is increasing.
In this paper we study the existence of solutions $u\in H^1(\R^N)$ forthe problem $-\Delta u+a(x)u=|u|^{p-2}u$, where $N\ge 2$ and $p$ issuperlinear and subcritical.The potential $a(x)\in L^\infty(\R^N)$ is such that $a(x)\ge c>0$ butis 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 adirection exists along which the potential is increasing.
2010, 28(2): 845-863
doi: 10.3934/dcds.2010.28.845
+[Abstract](2530)
+[PDF](252.0KB)
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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 $+\infty$.
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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