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作者：Bernard, Patrick; Contreras, Gonzalo
摘要：We prove that a generic Lagrangian has finitely many minimizing measures for every cohomology class.
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作者：Trudinger, Neil S.; Wang, Xu-Jia
摘要：In this paper, we prove global second derivative estimates for solutions of the Dirichlet problem for the Monge-Ampere equation when the inhomogeneous term is only assumed to be Holder continuous. As a consequence of our approach, we also establish the existence and uniqueness of globally smooth solutions to the second boundary value problem for the affine maximal surface equation and affine mean curvature equation.
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作者：Viana, Marcelo
摘要：We prove that for any s > 0 the majority of C-s linear cocycles over any hyperbolic (uniformly or not) ergodic transformation exhibit some nonzero Lyapunov exponent: this is true for an open dense subset of cocycles and, actually, vanishing Lyapunov exponents correspond to codimension-infinity. This open dense subset is described in terms of a geometric condition involving the behavior of the cocycle over certain heteroclinic orbits of the transformation.
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作者：Djadli, Zindine; Malchiodi, Andrea
摘要：Given a compact four dimensional manifold, we prove existence of conformal metrics with constant Q-curvature under generic assumptions. The problem amounts to solving a fourth-order nonlinear elliptic equation with variational structure. Since the corresponding Euler functional is in general unbounded from above and from below, we employ topological methods and min-max schemes, jointly with the compactness result of [35].
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作者：Farb, Benson; Weinberger, Shmuel
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作者：Phuc, Nguyen Cong; Verbitsky, Igor E.
摘要：The existence problem is solved, and global pointwise estimates of solutions are obtained for quasilinear and Hessian equations of Lane-Emden type, including the following two model problems: -Delta(p)u = u(q) + mu, F-k[-u] = u(q) + mu, u >= 0, on R-n, or on a bounded domain Omega subset of R-n. Here Delta(p) is the p-Laplacian defined by Delta(p)u = div (del u vertical bar del u vertical bar(p-2)), and F-k[u] is the k-Hessian defined as the sum of k x k principal minors of the Hessian matrix ...
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作者：Auroux, Denis; Katzarkov, Ludmil; Orlov, Dmitri
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作者：Chen, Kuo-Chang
摘要：Poincare made the first attempt in 1896 on applying variational calculus to the three-body problem and observed that collision orbits do not necessarily have higher values of action than classical solutions. Little progress had been made on resolving this difficulty until a recent breakthrough by Chenciner and Montgomery. Afterward, variational methods were successfully applied to the N-body problem to construct new classes of solutions. In order to avoid collisions, the problem is confined to...
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作者：Musin, Oleg R.
摘要：The kissing number problem asks for the maximal number k(n) of equal size nonoverlapping spheres in n-dimensional space that call touch another sphere of the same size. This problem in dimension three was the subject of a famous discussion between Isaac Newton and David Gregory in 1694. In three dimensions the problem was finally solved only in 1953 by Schutte and van der Waerden. In this paper we present a solution of a long-standing problem about the kissing number in four dimensions. Namely...
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作者：Colliander, J.; Keel, M.; Staffilani, G.; Takaoka, H.; Tao, T.
摘要：We obtain global well-posedness, scattering, and global L-t,x(10) spacetime bounds for energy-class solutions to the quintic defocusing Schrodinger equation in R1+3, which is energy-critical. In particular, this establishes global existence of classical solutions. Our work extends the results of Bourgain [4] and Grillakis [20], which handled the radial case. The method is similar in spirit; to the induction-on-energy strategy of Bourgain [4), but we perform the induction analysis in both frequ...
