Tristan Rivière

Vortices for a variational problem related to superconductivity

Abstract We study minimizers of Ginzburg-Landau functional, which depend on a parameter ϵ. These functional appear in superconductivity and two dimensional abelian Higgs models. We study the asymptotic limit, as ϵ → 0, of minimizers and show that the limiting configuration has vortices, which have topological degree one.

Conservation laws for fourth order systems in four dimensions

Following an approach of the second author (Rivière, 2007) for conformally invariant variational problems in two dimensions, we show in four dimensions the existence of a conservation law for fourth order systems, which includes both intrinsic and extrinsic biharmonic maps. With the help of this conservation law we prove the continuity of weak solutions of this system. Moreover we use the conservation law to derive the existence of a unique global weak solution of the extrinsic biharmonic map flow in the energy space.

Vortex energy and vortex bending for a rotating Bose-Einstein condensate

For a Bose-Einstein condensate placed in a rotating trap, we give a simplified expression of the Gross-Pitaevskii energy in the Thomas Fermi regime, which only depends on the number and shape of the vortex lines. Then we check numerically that when there is one vortex line, our simplified expression leads to solutions with a bent vortex for a range of rotationnal velocities and trap parameters which are consistent with the experiments.

Compactness, Kinetic Formulation, and Entropies for a Problem Related to Micromagnetics

Abstract We carry on the study of (Rivière T, Serfaty S. Limiting domain wall energy for a problem related to micromagnetics. Comm Pure Appl Math 2001; 54(3):294–338.) on the asymptotics of a family of energy-functionals related to micromagnetics. We prove compactness for families of uniformly bounded energies releasing the LBP condition we had previously set. Such families converge to unit-valued divergence-free vector-fields that are tangent to the boundary of the domain, and we found in (Rivière T, Serfaty S. Limiting domain wall energy for a problem related to micromagnetics. Comm Pure Appl Math 2001; 54(3):294–338.) that the energy-functionals Γ-converge to a limiting jump-energy of such configurations. We examine the behavior of certain truncated fields which serve to construct “entropies,” and to provide an improved lower bound. We give a kinetic formulation of the problem, and show that the limiting divergence-free problem is supplemented, in the case of minimizers, with a sign condition which can in turn, using the kinetic formulation, be interpreted as an entropy condition that plays a role in uniqueness questions.

Energy quantization for harmonic maps

In this paper we establish the higher-dimensional energy bubbling results for harmonic maps to spheres. We have shown in particular that the energy density of concentrations has to be the sum of energies of harmonic maps from the standard 2dimensional spheres. The result also applies to the structure of tangent maps of stationary harmonic maps at either a singularity or infinity. 0. Introduction Let M , N be smooth, compact Riemannian manifolds without boundary. Suppose u : M → N is a smooth harmonic map such that the homotopy class, [u], of u is not trivial. Then it follows easily from the small energy regularity theorem of R. Schoen and K. Uhlenbeck (cf. [ Sc]) that the total energy of the map u is

Resolutions of the prescribed volume form equation

For a given volume formfdx on a bounded regular domain Ω in IRn, we are looking for a transformationu of Ω, keeping the boundary fixed and which sends the Lebesgue measuredx intofdx (i.e. we solve det (Δu)=f. Forf in various spaces, we propose two different constructions which ensure the existence ofu with some gain of regularity. Our methods permit the recovery Dacorogna and Mosers results [4], but also, we prove the existence of suchu in Hölder spaces forf inC0, or even inL∞.

Dense Subsets of H1/2(S2, S1)

We prove that the maps from S2 intoS1 having a finite number of isolated singularities ofdegree ±1 are dense for the strong topology inH1/2(S2, S1). We also prove that smooth maps are densein H1/2(S2, S1)for the sequentially weak topology andthat this is no more the case in Hs(S2, S1) for s> 1/2.

Lipschitz conformal immersions from degenerating Riemann surfaces with L2-boundedsecond fundamental forms

Abstract. We give an asymptotic lower bound for the Willmore energy of weak immersions with degenerating conformal class. This lower bound is used in several other works. It is for instance one of the ingredients used by the author (2010) for providing an alternative proof of the one by L. Simon of the existence of a smooth torus minimizing the Willmore energy. The main result of the present paper has been independently obtained by Kuwert and Li (2010).

A Sharp Nonlinear Gagliardo-Nirenberg-Type Estimate and Applications to the Regularity of Elliptic Systems

Abstract We prove the inequality and to give a sample of possible applications, we show how it can be used to obtain ϵ-regularity results for the solutions of a wide class of nonlinear degenerate elliptic systems where G grows as |∇u| p .

The rectifiability of entropy measures in one space dimension

Abstract We show that entropy solutions to 1-dimensional scalar conservation laws for totally nonlinear fluxes and for arbitrary measurable bounded data have a structure similar to the one of BV maps without being always BV. The singular set—shock waves—of such solutions is contained in a countable union of C1 curves and H 1 almost everywhere along these curves the solution has left and right approximate limits. The entropy production is concentrated on the shock waves and can be explicitly computed in terms of the approximate limits. The solution is approximately continuous H 1 almost everywhere outside this union of curves.