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Doubling constant mean curvature tori in S 3

Adrian ButscherFrank Pacard — 2006

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

The Clifford tori in S 3 constitute a one-parameter family of flat, two-dimensional, constant mean curvature (CMC) submanifolds. This paper demonstrates that new, topologically non-trivial CMC surfaces resembling a pair of neighbouring Clifford tori connected at a sub-lattice consisting of at least two points by small catenoidal bridges can be constructed by perturbative PDE methods. That is, one can create a submanifold that has almost everywhere constant mean curvature by gluing a re-scaled catenoid...

Multiple end solutions to the Allen-Cahn equation in 2

Michał KowalczykYong LiuFrank Pacard

Séminaire Laurent Schwartz — EDP et applications

An entire solution of the Allen-Cahn equation Δ u = f ( u ) , where f is an odd function and has exactly three zeros at ± 1 and 0 , e.g. f ( u ) = u ( u 2 - 1 ) , is called a 2 k end solution if its nodal set is asymptotic to 2 k half lines, and if along each of these half lines the function u looks (up to a multiplication by - 1 ) like the one dimensional, odd, heteroclinic solution H , of H ' ' = f ( H ) . In this paper we present some recent advances in the theory of the multiple end solutions. We begin with the description of the moduli space of such solutions....

Finite-energy sign-changing solutions with dihedral symmetry for the stationary nonlinear Schrödinger equation

Monica MussoFrank PacardJuncheng Wei — 2012

Journal of the European Mathematical Society

We address the problem of the existence of finite energy solitary waves for nonlinear Klein-Gordon or Schrödinger type equations Δ u - u + f ( u ) = 0 in N , u H 1 ( N ) , where N 2 . Under natural conditions on the nonlinearity f , we prove the existence of 𝑖𝑛𝑓𝑖𝑛𝑖𝑡𝑒𝑙𝑦𝑚𝑎𝑛𝑦𝑛𝑜𝑛𝑟𝑎𝑑𝑖𝑎𝑙𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛𝑠 in any dimension N 2 . Our result complements earlier works of Bartsch and Willem ( N = 4 𝚘𝚛 N 6 ) and Lorca-Ubilla ( N = 5 ) where solutions invariant under the action of O ( 2 ) × O ( N - 2 ) are constructed. In contrast, the solutions we construct are invariant under the action of D k × O ( N - 2 ) where D k O ( 2 ) denotes the dihedral group...

Bubbling along boundary geodesics near the second critical exponent

Manuel del PinoMonica MussoFrank Pacard — 2010

Journal of the European Mathematical Society

The role of the second critical exponent p = ( n + 1 ) / ( n - 3 ) , the Sobolev critical exponent in one dimension less, is investigated for the classical Lane–Emden–Fowler problem Δ u + u p = 0 , u > 0 under zero Dirichlet boundary conditions, in a domain Ω in n with bounded, smooth boundary. Given Γ , a geodesic of the boundary with negative inner normal curvature we find that for p = ( n + 1 ) / ( n - 3 - ε ) , there exists a solution u ε such that | u ε | 2 converges weakly to a Dirac measure on Γ as ε 0 + , provided that Γ is nondegenerate in the sense of second variations of...

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