Heat flow for p-harmonic maps with small initial data

We study developing singularities for surfaces of rotation with free boundaries and evolving under volume-preserving mean curvature flow. We show that singularities form a finite, discrete set along the axis of rotation. We prove a monotonicity formula and conclude that type I singularities are asymtotically cylindrical.     

Uniqueness of harmonic maps with small energy

Harmonic maps from B ₁ (0, ℝ³) to a smooth compact target manifold N with uniformly small scaled energy (see assumption (2) below) are shown to be unique for their boundary values. Received: 12 May 1997     

Harmonic Hopf constructions between spheres II

Calculus of Variations and Partial Differential Equations -     

A method to exclude branch points of minimal surfaces

The central problem of this paper is to exclude boundary branch points of minimal surfaces. The method consists in showing that the third derivative of the Dirichlet energy is negative along well-chosen paths in admissible Jacobi field directions, if a “Schüffler condition” is satisfied. Received July 21, 1997 / Accepted October 3, 1997     

Relative isoperimetric inequality and¶linear isoperimetric inequality for minimal submanifolds

We prove an optimal relative isoperimetric inequality

A characterization of graphs which can be approximated in area by smooth graphs

For vector valued maps, convergence in W ^1,1 and of all minors of the Jacobian matrix in L ¹ is equivalent to convergence weakly in the sense of currents and in area for graphs. We show that maps defined on domains of dimension n≥ 3 can be approximated strongly in this sense by smooth maps if and only if the same property holds for the restriction to a.e. 2-dimensional plane intersecting the domain. Received April 29, 1999 / final version received July 21, 2000?Published online September 25, 2000     

A note on the Hodge theory for functionals with linear growth

We study the Hodge decomposition of L ¹-(and measure-) differential forms over a compact manifold without boundary, giving positive results and counterexamples. The theory is then applied to the relaxation and minimization, in cohomology classes, of convex functionals with linear growth. This corresponds to a non-linear version of the Hodge theory, in the spirit of L. M. Sibner and R. J. Sibner [SS]. Received: 19 November 1997 / Revised version: 18 May 1998     

Uniform partial regularity of quasi minimizers for the perimeter

Quasi minimizers for the perimeter are measurable subsets G of such that for all variations of G with and for a given increasing function such that . We prove here that, given , G a reduced quasi minimizer, and , there are , with , and , homeomorphic to a closed ball with radius t in , such that for some absolute constant . The constant above depends only on n, and . If moreover for some , we prove that we can find such a ball such that is a dimensional graph of class . This will be obtained proving that a quasi minimizer is equivalent to some set which satisfies the condition B. This condition gives some kind of uniform control on the flatness of the boundary and then criterions proven by Ambrosio-Paolini and Tamanini can be applied to get the required regularity properties. Received: July 12, 1999 / Accepted: October 1, 1999     

The energy of Hopf vector fields

The 3-dimensional Hopf vector field is shown to be a stable harmonic section of the unit tangent bundle. In contrast, higher dimensional Hopf vector fields are unstable harmonic sections; indeed, there is a natural variation through smooth unit vector fields which is locally energy-decreasing, and whose asymptotic limit is a singular vector field of finite energy. This energy is explicitly calculated, and conjectured to be the infimum of the energy functional over all smooth unit vector fields. Received: 17 March 1999     

KAM theorem of symplectic algorithms for Hamiltonian systems

Summary. In this paper we prove that an analog of the celebrated KAM theorem holds for symplectic algorithms, which Channel and Scovel (1990), Feng Kang (1991) and Sanz-Serna and Calvo (1994) suggested a few years ago. The main results consist of the existence of invariant tori, with a smooth foliation structure, of a symplectic numerical algorithm when it applies to a generic integrable Hamiltonian system if the system is analytic and the time-step size of the algorithm is s ufficiently small. This existence result also implies that the algorithm, when it is applied to a generic integrable system, possesses n independent smooth invariant functions which are in involution and well-defined on the set filled by the invariant tori in the sense of Whitney. The invariant tori are just the level sets of these functions. Some quantitative results about the numerical invariant tori of the algorithm approximating the exact ones of the system are also given. Received December 27, 1997 / Revised version received July 15, 1998 / Published online: July 7, 1999     

Multiplicity of homoclinics for a class of time recurrent second order Hamiltonian systems

We prove the existence of infinitely many homoclinic solutions for a class of second order Hamiltonian systems in of the form , where we assume the existence of a sequence such that and as for any . Moreover, under a suitable non degeneracy condition, we prove that this class of systems admits multibump solutions. Received February 2, 1996 / In revised form July 5, 1996 / Accepted October 10, 1996     

Harmonic morphisms as unit normal bundles¶of minimal surfaces

Let be an isometric immersion between Riemannian manifolds and be the unit normal bundle of f. We discuss two natural Riemannian metrics on the total space and necessary and sufficient conditions on f for the projection map to be a harmonic morphism. We show that the projection map of the unit normal bundle of a minimal surface in a Riemannian manifold is a harmonic morphism with totally geodesic fibres. Received: 6 February 1999     

On the existence of H-surfaces into Riemannian manifolds

This paper considers the existence of a local minimizer of a conformally invariant functional defined on a space of maps of a closed Riemann surface into a compact Riemannian manifold . The functional is defined for a given tensor on of type (1,2) and we call its extremal an -surface. In fact, we prove that there exists a local minimizer of the functional in a given homotopy class under certain conditions on , and the minimum of the Dirichlet integral of maps of the homotopy class. Received January 21, 1994 / Received in revised form October 24, 1995 / Accepted March 15, 1996     

Bauer's maximum principle and hulls of sets

We use the Bauer maximum principle for quasiconvex, polyconvex and rank-one convex functions to derive Krein-Milman-type theorems for compact sets in . Further we show that in general the set of quasiconvex extreme points is not invariant under transposition and it is different from the set of rank-one convex extreme points. Finally, a set in with different polyconvex, quasiconvex and rank-one convex hulls is constructed. Received September 14, 1999 / Accepted January 14, 2000 /Published online July 20, 2000     

Limiting angles of Γ-martingales

Suppose that M is a complete, simply connected Riemannian manifold of non-positive sectional curvature with dimension m≥ 3 and that, outside a fixed compact set, the sectional curvatures are bounded above by −c ₁/{r ² ln r} and below by −c ₂ r ², where c ₁ and c ₂ are two positive constants and r is the geodesic distance from a fixed point. We show that, when κ≥ 1 satisfies certain conditions, the angular part of a κ-quasi-conformal Γ-martingale on M tends to a limit as time tends to infinity and the closure of the support of the distribution of this limit is the entire sphere at infinity. This improves both a result of Le for Brownian motion and also results concerning the non-existence of κ-quasi-conformal harmonic maps from certain types of Riemannian manifolds into M. Received: 19 September 1997