101,736 research outputs found
Estimates and computations in Rabinowitz-Floer homology
No full textThe Rabinowitz-Floer homology of a Liouville domain W is the Floer homology of the Rabinowitz free period Hamiltonian action functional associated to a Hamiltonian whose zero energy level is the boundary of W. This invariant has been introduced by K. Cieliebak and U. Frauenfelder and has already found several applications in symplectic topology and in Hamiltonian dynamics. Together with A. Oancea, the same authors have recently computed the Rabinowitz-Floer homology of the cotangent disk bundle D^*M of a closed Riemannian manifold M, by means of an exact sequence relating the Rabinowitz-Floer homology of D^*M with its symplectic homology and cohomology. The first aim of this paper is to present a chain level construction of this exact sequence. In fact, we show that this sequence is the long homology sequence induced by a short exact sequence of chain complexes, which involves the Morse chain complex and the Morse differential complex of the energy functional for closed geodesics on M. These chain maps are defined by considering spaces of solutions of the Rabinowitz-Floer equation on half-cylinders, with suitable boundary conditions which couple them with the negative gradient flow of the geodesic energy functional.
The second aim is to generalize this construction to the case of a fiberwise uniformly convex compact subset W of T^*M whose interior part contains a Lagrangian graph. Equivalently, W is the energy sublevel associated to an arbitrary Tonelli Lagrangian L on TM and to any energy level which is larger than the strict Mane' critical value of . In this case, the energy functional for closed geodesics is replaced by the free period Lagrangian action functional associated to a suitable calibration of L. An important issue in our analysis is to extend the uniform estimates for the solutions of the Rabinowitz-Floer equation - both on cylinders and on half-cylinders - to Hamiltonians which have quadratic growth in the momenta. These uniform estimates are obtained by the Aleksandrov integral version of the maximum principle. In the case of half-cylinders, they are obtained by an Aleksandrov-type maximum principle with Neumann conditions on part of the boundary
Heteroclinic solutions between stationary points at different energy levels
No full textConsider the system of equations
The main goal of this paper is to present a simple minimization method
to find heteroclinic connections between isolated critical points of
, say and , which are local maxima but do not necessarily
have the same value of . In particular we prove that there exist
heteroclinic solutions from to and from to for a
class of positive slowly oscillating periodic functions provided
is sufficiently small (and another
technical condition is satisfied). Note that when ,
cannot be constant be conservation of energy. Existence of
``multi-bump'' solutions is also proved
Letter, [Author unclear] to Paulina T. Merritt
No full textHandwritten letter to Paulina Merritt from an unknown author, October 1, 1876.
Periodic solutions for Hamiltonian systems without Ambrosetti–Rabinowitz condition and spectrum 0
No full textAbstractIn this paper, we consider the superquadratic second order Hamiltonian systemu″(t)+A(t)u(t)+∇H(t,u(t))=0,t∈R. Our main results here allow the classical Ambrosetti–Rabinowitz superlinear condition to be replaced by a general superquadratic condition, and 0 lies in a gap of σ(B), where B:=−d2dt2−A(t). We will study the ground state periodic solutions for this problem. The main idea here lies in an application of a variant generalized weak linking theorem for strongly indefinite problem developed by Schechter and Zou
Homoclinic solutions for second-order non-autonomous Hamiltonian systems without global Ambrosetti-Rabinowitz conditions
No full textThis article studies the existence of homoclinic solutions for the second-order non-autonomous Hamiltonian system
where is a symmetric and positive definite matrix for all . The function is not assumed to satisfy the global Ambrosetti-Rabinowitz condition. Assuming reasonable conditions on and , we prove the existence of at least one nontrivial homoclinic solution, and for even in , we prove the existence of infinitely many homoclinic solutions.Mathematic
Homoclinic solutions for second-order non-autonomous Hamiltonian systems without global Ambrosetti-Rabinowitz conditions
No full textThis article studies the existence of homoclinic solutions for the second-order non-autonomous Hamiltonian system where is a symmetric and positive definite matrix for all . The function is not assumed to satisfy the global Ambrosetti-Rabinowitz condition. Assuming reasonable conditions on and , we prove the existence of at least one nontrivial homoclinic solution, and for even in , we prove the existence of infinitely many homoclinic solutions
Crustal structure in the Western Somali Basin
No full textAs part of integrated marine geophysical studies in the Western Somali Basin, we performed 118 sonobuoy experiments to define better the crustal structure of the margins and basin created by the separation of Madagascar and Africa. After using T2/X2, conventional slope-intercept methods, and slant-stacked t-p techniques to analyse the data, we combined our solutions with all previous velocity information for the area. Velocity functions were derived for the sediment column, and we detected a high-velocity (4.58 ± 0.29 km s–1) sediment layer overlying acoustic basement. We confirmed that the crust is indeed seismically oceanic, and that it may be considered either in terms of a layered model – layers 2B (5.42 ± 0.19 km s–1), 2C (6.23 ± 0.22 km s–1), 3 (7.03 ± 0.25 km s–1), and mantle (7.85 ± 0.32 km s–1) were identified – or a more complex gradient model in which layer 2 is marked by a steeper velocity gradient than underlying layer 3. Integrated igneous crustal thicknesses (1.62 ± 0.22 s, 5.22 ± 0.64 km) are significantly less than what is considered normal. We present a revised seismic transect across the East African margin, as well as total sediment thickness, depth to basement and crustal thickness maps.<br/
Multiplicity of positive solutions for a fractional p&q-Laplacian problem in RN
No full textIn this paper we deal with the following fractional p&q-Laplacian problem: {(−Δ)psu+(−Δ)qsu+V(εx)(|u|p−2u+|u|q−2u)=f(u) in RN,u∈Ws,p(RN)∩Ws,q(RN),u>0 in RN, where s∈(0,1), ε>0 is a small parameter, [Formula presented], (−Δ)ts, with t∈{p,q}, is the fractional (s,t)-Laplacian operator, V:RN→R is a continuous function satisfying the global Rabinowitz condition, and f:R→R is a continuous function with subcritical growth. Using suitable variational arguments and Ljusternik-Schnirelmann category theory, we prove that the above problem admits multiple solutions for ε>0 small enough
Handwritten biographical information on Paulina T. McClung Merritt
No full textA handwritten biography of Paulina T. McClung Merritt by an unknown author, 1892.
Heterogeneous and tissue-specific regulation of effector T cell responses by IFN-gamma during Plasmodium berghei ANKA infection.
No full textIFN-γ and T cells are both required for the development of experimental cerebral malaria during Plasmodium berghei ANKA infection. Surprisingly, however, the role of IFN-γ in shaping the effector CD4(+) and CD8(+) T cell response during this infection has not been examined in detail. To address this, we have compared the effector T cell responses in wild-type and IFN-γ(-/-) mice during P. berghei ANKA infection. The expansion of splenic CD4(+) and CD8(+) T cells during P. berghei ANKA infection was unaffected by the absence of IFN-γ, but the contraction phase of the T cell response was significantly attenuated. Splenic T cell activation and effector function were essentially normal in IFN-γ(-/-) mice; however, the migration to, and accumulation of, effector CD4(+) and CD8(+) T cells in the lung, liver, and brain was altered in IFN-γ(-/-) mice. Interestingly, activation and accumulation of T cells in various nonlymphoid organs was differently affected by lack of IFN-γ, suggesting that IFN-γ influences T cell effector function to varying levels in different anatomical locations. Importantly, control of splenic T cell numbers during P. berghei ANKA infection depended on active IFN-γ-dependent environmental signals--leading to T cell apoptosis--rather than upon intrinsic alterations in T cell programming. To our knowledge, this is the first study to fully investigate the role of IFN-γ in modulating T cell function during P. berghei ANKA infection and reveals that IFN-γ is required for efficient contraction of the pool of activated T cells
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