The Willmore boundary problem

We consider the Euler–Lagrange equation of the Willmore functional coupled with Dirichlet and Neumann boundary conditions on a given curve. We prove existence of a branched solution, and for Willmore energy < 8π, we prove the existence of a smooth, embedded solution.     

Local Solution and Extension to the Calabi Flow

We consider the local solution of the Calabi flow for rough initial data. In particular, we prove that for any smooth metric, there is a C ^α neighborhood such that the Calabi flow has a short time solution for any C ^α metric in the neighborhood. We also prove that on a compact Kähler surface, if the evolving metrics of the Calabi flow are all L ^∞ equivalent, then the Calabi flow exists for all time and converges to an extremal metric subsequently.     

On ϵ-Collocation for Pseudodifferential Equations on a Closed Curve

This paper is devoted to the approximate solution of one-dimensional pseudodifferential equations on a closed curve via spline collocation methods with variable collocation points and represents a continuation of [11]. We give necessary and sufficient conditions ensuring the L²-convergence for operators with smooth and piecewise continuous coefficients.     

Gradient Flow for the Helfrich Functional

The author studies the L ² gradient flow of the Helfrich functional, which is a functional describing the shapes of human red blood cells. For any ?? _i ?? 0 and c ₀, the author obtains a lower bound on the lifespan of the smooth solution, which depends only on the concentration of curvature for the initial surface.     

Morse theory for the space of Higgs <Emphasis Type="Italic">G</Emphasis>–bundles

Fix a C ^∞ principal G–bundle E⁰_G{E^0_G} on a compact connected Riemann surface X, where G is a connected complex reductive linear algebraic group. We consider the gradient flow of the Yang–Mills–Higgs functional on the cotangent bundle of the space of all smooth connections on E⁰_G{E^0_G}. We prove that this flow preserves the subset of Higgs G–bundles, and, furthermore, the flow emanating from any point of this subset has a limit. Given a Higgs G–bundle, we identify the limit point of the integral curve passing through it. These generalize the results of the second named author on Higgs vector bundles.     

Schrödinger flow of maps into symplectic manifolds

The definition of Schrödinger flow is proposed. It is indicated that the flow of ferromagnetic chain is actually Schrödinger flow of maps intoS ², and that there exists a unique local smooth solution for the initial value problem of one-dimensional Schrödinger flow of maps into Kahler manifolds. In case the targets are Kähler manifolds with constant curvature, it is proved that one-dimensional Schrödinger flow admits a unique global smooth solution.     

Constructing C3 shape preserving interpolating space curves

We present a method for constructing shape-preserving C ³ interpolants in R ³. The resulting curve is obtained by adding a polynomial perturbation of high degree to a curve which is shape-preserving but not sufficiently smooth. The degree of the perturbed curve is selected in order to maintain the shape properties of the basic curve.     

On Cremona transformations and quadratic complexes

We study the relation between Cremona transformations in space and quadratic line complexes. We show that it is possible to associate a space Cremona transformation to each smooth quadratic line complex once we choose two distinct lines contained in the complex. Such Cremona transformations are cubo-cubic and we classify them in terms of the relative position of the lines chosen. It turns out that the base locus of such a transformation contains a smooth genus two quintic curve. Conversely, we show that given a smooth quintic curve C of genus 2 in ℙ³ every Cremona transformation containing C in its base locus factorizes through a smooth quadratic line complex as before. We consider also some cases where the curve C is singular, and we give examples both when the quadratic line complex is smooth and singular.     

Rotationally symmetric 1-harmonic maps from D 2 to S 2

We consider rotationally symmetric 1-harmonic maps from D ² to S ² subject to Dirichlet boundary conditions. We prove that the corresponding energy—a degenerate non-convex functional with linear growth—admits a unique minimizer, and that the minimizer is smooth in the bulk and continuously differentiable up to the boundary. We also show that, in contrast with 2-harmonic maps, a range of boundary data exists such that the energy admits more than one smooth critical point: more precisely, we prove that the corresponding Euler–Lagrange equation admits a unique (up to scaling and symmetries) global solution, which turns out to be oscillating, and we characterize the minimizer and the smooth critical points of the energy as the monotone, respectively non-monotone, branches of such solution. R. Dal Passo passed away on 8th August 2007. Endowed with great strength, creativity and humanity, Roberta has been an outstanding mathematician, an extraordinary teacher and a wonderful friend. Farewell, Roberta.     

On the curve diffusion flow of closed plane curves

In this paper, we consider the steepest descent H ^?1-gradient flow of the length functional for immersed plane curves, known as the curve diffusion flow. It is known that under this flow there exist both initially immersed curves that develop at least one singularity in finite time and initially embedded curves that self-intersect in finite time. We prove that under the flow closed curves with initial data close to a round circle in the sense of normalised L ² oscillation of curvature exist for all time and converge exponentially fast to a round circle. This implies that for a sufficiently large ‘waiting time’, the evolving curves are strictly convex. We provide an optimal estimate for this waiting time, which gives a quantified feeling for the magnitude to which the maximum principle fails. We are also able to control the maximum of the multiplicity of the curve along the evolution. A corollary of this estimate is that initially embedded curves satisfying the hypotheses of the global existence theorem remain embedded. Finally, as an application we obtain a rigidity statement for closed planar curves with winding number one.     

Lagrangian mean curvature flow for entire Lipschitz graphs

We consider the mean curvature flow of entire Lagrangian graphs with Lipschitz continuous initial data. Assuming only a certain bound on the Lipschitz norm of an initial entire Lagrangian graph in ${{\mathbb R}^{2n}}$ , we show that the parabolic Eq. 1.1 has a longtime solution which is smooth for all positive time and satisfies uniform estimates away from time t?=?0. In particular, under the mean curvature flow (1.2) the graph immediately becomes smooth and the solution exists for all time such that the second fundamental form decays uniformly to 0 on the graph as t → ∞. Our assumption on the Lipschitz norm is equivalent to the underlying Lagrangian potential u being uniformly convex with its Hessian bounded in L ^∞. As an application of this result we provide conditions under which an entire Lipschitz Lagrangian graph converges after rescaling to a self-expanding solution to the mean curvature flow.     

Shape Optimization for Semi-Linear Elliptic Equations Based on an Embedding Domain Method

We study a class of shape optimization problems for semi-linear elliptic equations with Dirichlet boundary conditions in smooth domains in ℝ². A part of the boundary of the domain is variable as the graph of a smooth function. The problem is equivalently reformulated on a fixed domain. Continuity of the solution to the state equation with respect to domain variations is shown. This is used to obtain differentiability in the general case, and moreover a useful formula for the gradient of the cost functional in the case where the principal part of the differential operator is the Laplacian. Online publication 23 January 2004.     

Oil displacement by water and gas in a porous medium: the Riemann problem

In this work we present the construction of the Riemann solution for a system of two conservation laws representing displacement in immiscible three-phase flow. The porousmedium is initially filled with oil and small amounts of water and gas; then a fixed proportion of water and gas is injected. We use the wave curve method to determine the wave sequences in the Riemann solution for arbitrary initial and injection data in the above mentioned class. We show the L_Loc¹-stability of the Riemann solution with variation of data. We do not verify uniqueness of the Riemann solution, but we believe that it is valid.     

Curve Shortening Flow in Arbitrary Dimensional Euclidian Space

In this paper, curve shortening flow in Euclidian space R^n（n≥3） is studied, and S. Altschuler＇s results about flow for space curves are generalized. We prove that the curve shortening flow converges to a straight line in infinite time if the initial curve is a ramp. We also prove the planar phenomenon when the curve shortening flow blows up.     

On the existence of incompressible current–vortex sheets: study of a linearized free boundary value problem

We study the initial boundary value problem resulting from the linearization of the equations of ideal incompressible magnetohydrodynamics and the jump conditions on the hypersurface of tangential discontinuity (current–vortex sheet) about an unsteady piecewise smooth solution. Under some assumptions on the unperturbed flow, we prove an energy a priori estimate for the linearized problem. Since the so‐called loss of derivatives in the normal direction to the boundary takes place even for the constant coefficients linearized problem, for the variable coefficients problem and non‐planar current–vortex sheets the natural functional setting is provided by the anisotropic weighted Sobolev space W₂^1,σ. The result of this paper is a necessary step to prove the local in time existence of solutions of the original non‐linear free boundary value problem. The uniqueness of the regular solution of this problem follows already from the a priori estimate we obtain for the linearized problem. Copyright © 2005 John Wiley & Sons, Ltd.     

Flows on open manifolds with positively Lagrange stable trajectories

We give the conditions for a flow generated by a smooth vector field X which guarantee that every smooth vectorfield Y in some C⁰-neighborhood of X defines a flow with positively Lagrange stable trajectories.     

A reformulation of Le's conjecture

We give a new proof of Le's conjecture on surface germs in ?³ having as link a topological sphere for the case of surface singularities containing a smooth curve. Our proof leads to a reformulation of the general case of the conjecture into a problem of plane curve singularities and their relative polar curves.     

On a length preserving curve flow

In this paper, we consider a new length preserving curve flow for closed convex curves in the plane. We show that the flow exists globally, the area of the region bounded by the evolving curve is increasing, and the evolving curve converges to the circle in C ^?? topology as t ?? ??.     

Global nonautonomous Schr?dinger flows on Hermitian locally symmetric spaces

In this paper, we consider the global existence of one-dimensional nonautonomous (inhomogeneous) Schr?dinger flow. By exploiting geometric symmetries, we prove that, given a smooth initial map, the Cauchy problem of the nonautonomous (inhomogeneous) Schr?dinger flow from S¹ into a Hermitian locally symmetric space admits a unique global smooth solution, and then we address the global existence of the Cauchy problem of inhomogeneous Heisenberg spin ferromagnet system.