Patricia Bauman

Positive solutions of elliptic equations in nondivergence form and their adjoints

On considere des operateurs uniformement elliptiques de la forme L=Σ i,j=1 n a ij (X)•D XiXj 2 +Σ i=1 n b i (X)•D Xi a coefficients bornes mesurables de R n . On demontre un theoreme de comparaison pour des solutions positives de Lu=0 dans un domaine de Lipschitz borne D de R n

A nonconvex variational problem related to change of phase

We investigate the elastostatic deformation of a tube whose crosssection is a convex ring Ω. The outer lateral surface is assumed to be held fixed and the inner surface is displaced in the axial direction a uniform distanceh. The problem becomes one of seeking minimizers for a functionalJ(u) = ∫Ωω(|∇u|) dx whereu(x) is the axial displacement andω(·) is nonconvex. When Ω is an annulus minimizers are known to exist. We prove existence and nonexistence results by studying a relaxed problem obtained by replacingω(|·|) with its lower convex envelope,ω**(|·|). If a minimizer forJ(·) exists it is also a solution to the relaxed problem and this leads to an overdetermined problem in some cases.WhenJ(·) has no minimizer, solutions of the relaxed problem are of interest. We show that the relaxed problem has a unique solution and give detailed information on its structure.

Maximum principles and a priori estimates for a class of problems from nonlinear elasticity

Abstract We consider smooth solutions, , to the nonlinear elliptic system associated with a two dimensional elastic material which has energy functional The function H( d ) is nonnegative, convex and unbounded in a neighborhood of zero. Two maximum principles are proved for and we show that if Ωʹ ⊂ ⊂ Ω then and are bounded a priori in terms of and for some p = p ( H ).

Analysis of Nematic Liquid Crystals with Disclination Lines

We investigate the structure of nematic liquid crystal thin films described by the Landau–de Gennes tensor-valued order parameter model with Dirichlet boundary conditions on the sides of nonzero degree. We prove that as the elasticity constant goes to zero in the energy, a limiting uniaxial nematic texture forms with a finite number of defects, all of degree

Maximal smoothness of solutions to certain Euler-Lagrange equations from nonlinear elasticity

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Classical solutions to the time-dependent Ginzburg–Landau equations for a bounded superconducting body in a vacuum

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Analysis of solutions to the lawrence-doniach system for layered superconductors

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