Highly degenerate harmonic mean curvature flow

We study the evolution of a weakly convex surface in with flat sides by the Harmonic Mean Curvature flow. We establish the short time existence as well as the optimal regularity of the surface and we show that the boundaries of the flat sides evolve by the curve shortening flow. It follows from our results that a weakly convex surface with flat sides of class C ^k,γ , for some and 0  <  γ ≤ 1, remains in the same class under the flow. This distinguishes this flow from other, previously studied, degenerate parabolic equations, including the porous medium equation and the Gauss curvature flow with flat sides, where the regularity of the solution for t  >  0 does not depend on the regularity of the initial data. M. C. Caputo partially supported by the NSF grant DMS-03-54639. P. Daskalopoulos partially supported by the NSF grants DMS-01-02252, DMS-03-54639 and the EPSRC in the UK.     

Forced convex mean curvature flow in Euclidean spaces

We show the mean curvature flow of convex hypersurfaces in Euclidean spaces with a general forcing term may shrink to a point in finite time if the forcing term is small, or exist for all times and expand to infinity if the forcing term is large enough. The flow can converge to a round sphere in special cases. Long time existence and convergence of the normalization of the flow are studied.     

On surfaces of prescribed weighted mean curvature

Utilising a weight matrix we study surfaces of prescribed weighted mean curvature which yield a natural generalisation to critical points of anisotropic surface energies. We first derive a differential equation for the normal of immersions with prescribed weighted mean curvature, generalising a result of Clarenz and von der Mosel. Next we study graphs of prescribed weighted mean curvature, for which a quasilinear elliptic equation is proved. Using this equation, we can show height and boundary gradient estimates. Finally, we solve the Dirichlet problem for graphs of prescribed weighted mean curvature.     

Constant mean curvature spheres in Riemannian manifolds

We prove the existence of embedded spheres with large constant mean curvature in any compact Riemannian manifold (M, g). This result partially generalizes a result of R. Ye which handles the case where the scalar curvature function of the ambient manifold (M, g) has non-degenerate critical points.     

Equilibrium solutions to generalized motion by mean curvature

In this paper we consider viscosity equilibria to the mean curvature level set flow with a Dirichlet condition. The main result shows that almost every level set of an equilibrium solution is analytic off of a singular set of Hausdorff dimension at most n − 8 and that these level sets are stationary and stable with respect to the area functional. A key tool developed is a maximum principle for solutions to obstacle problems where the obstacle consists of (viscosity) minimal surfaces. Convergence to equilibrium as t → ∞ is also established for the associated initial-boundary value problem.     

Curvature estimates for graphs with prescribed mean curvature and flat normal bundle

We consider graphs with prescribed mean curvature and flat normal bundle. Using techniques of Schoen et al. (Acta Math 134:275–288, 1975) and Ecker and Huisken (Ann Inst H Poincaré Anal Non Linèaire 6:251–260, 1989), we derive the interior curvature estimate

Killing graphs with prescribed mean curvature

It is proved the existence and uniqueness of Killing graphs with prescribed mean curvature in a large class of Riemannian manifolds. M. Dajczer was partially supported by Procad, CNPq and Faperj. P. A. Hinojosa was partially supported by PADCT/CT-INFRA/CNPq/MCT Grant #620120/2004-5. J. H. de Lira was partially supported by CNPq and Funcap.     

A Faber-Krahn inequality for the Laplacian with Generalised Wentzell boundary conditions

We give an extension of the Faber-Krahn inequality to the Laplacian Δ on bounded Lipschitz domains , with generalised Wentzell boundary conditions on ∂Ω, where β, γ are nonzero real constants. We prove that when β, γ > 0, the ball B minimises the first eigenvalue with respect to all Lipschitz domains Ω of the same volume as B, and that B is the unique minimiser amongst C ²-domains. We also consider β, γ not both positive, and slightly extend what is known about the associated Wentzell operator and its resolvent in addition to considering an analogue of the Faber-Krahn inequality. This is based on the recent extension of the Faber-Krahn inequality to the Robin Laplacian. We also give a version of Cheeger’s inequality for the Wentzell Laplacian when β, γ > 0.      

Existence results for a class of porous medium type equations with a quadratic gradient term

In this paper we study the Dirichlet problem in Q _T = Ω × (0, T) for degenerate equations of porous medium-type with a lower order term:

Regularity results for quasilinear elliptic equations in the Heisenberg group

We prove regularity results for solutions to a class of quasilinear elliptic equations in divergence form in the Heisenberg group . The model case is the non-degenerate p-Laplacean operator where , and p is not too far from 2.     

Rearrangements and radial graphs of constant mean curvature in hyperbolic space

We investigate the problem of finding smooth hypersurfaces of constant mean curvature in hyperbolic space, which can be represented as radial graphs over a subdomain of the upper hemisphere. Our approach is variational and our main results are proved via rearrangement techniques. The second author was partially supported by NSF grant DMS 0603707.     

Ginzburg-Landau equation and motion by mean curvature, II: Development of the initial interface

In this paper, we study the short time behavior of the solutions of a sequence of Ginzburg-Landau equations indexed by ∈. We prove that under appropriate assumptions on the initial data, solutions converge to ±1 in short time and behave like the one-dimensional traveling wave across the interface. In particular, energy remains uniformly bounded in ∈. Partially supported by the NSF Grant DMS-9200801 and by the Army Research Office through the Center for Nonlinear Analysis.     

Local existence and finite time blow-up of solutions in the 2-D Keller-Segel system

We consider the 2-D Keller-Segel system (KS) for γ > 0. We first construct a mild solution of (KS) for every . The local existence time is characterized for with 1 < q _* < 2. Next, we prove the finite time blow-up of strong solution under the assumption and , where g(s) is an increasing function of s > 1 with an explicit representation. As an application of our mild solutions, an exact blow-up rate near the maximal existence time is obtained.      

Differential-geometric approach to the integrability of hydrodynamic chains: the Haantjes tensor

The integrability of an m-component system of hydrodynamic type, u _t = V(u)u _x , by the generalized hodograph method requires the diagonalizability of the m ×  m matrix V(u). This condition is known to be equivalent to the vanishing of the corresponding Haantjes tensor. We generalize this approach to hydrodynamic chains—infinite-component systems of hydrodynamic type for which the ∞ ×  ∞ matrix V(u) is ‘sufficiently sparse’. For such systems the Haantjes tensor is well-defined, and the calculation of its components involves finite summations only. We illustrate our approach by classifying broad classes of conservative and Hamiltonian hydrodynamic chains with the zero Haantjes tensor. We prove that the vanishing of the Haantjes tensor is a necessary condition for a hydrodynamic chain to possess an infinity of semi-Hamiltonian hydrodynamic reductions, thus providing an easy-to-verify necessary condition for the integrability.     

The mean curvature of cylindrically bounded submanifolds

We give an estimate of the mean curvature of a complete submanifold lying inside a closed cylinder in a product Riemannian manifold . It follows that a complete hypersurface of given constant mean curvature lying inside a closed circular cylinder in Euclidean space cannot be proper if the circular base is of sufficiently small radius. In particular, any possible counterexample to a conjecture of Calabi on complete minimal hypersurfaces cannot be proper. As another application of our method, we derive a result about the stochastic incompleteness of submanifolds with sufficiently small mean curvature. Dedicated to Professor Manfredo P. do Carmo on the occasion of his 80th birthday.     

Existence of type (II) regions and convexity and concavity of potential functionals corresponding to jumping nonlinear problems

In this paper we study the jumping nonlinear problem

Expanding convex immersed closed plane curves

We study the evolution driven by curvature of a given convex immersed closed plane curve. We show that it will converge to a self-similar solution eventually. This self-similar solution may or may not contain singularities. In case it does, we also have estimate on the curvature blow-up rate.     

The exponent for the mean square displacement of self-avoiding random walk on the Sierpinski gasket

The authors rigorously prove that the exponent for the mean square displacement of self-avoiding random walk on the Sierpinski gasket is

A parabolic almost monotonicity formula

We prove the parabolic counterpart of the almost monotonicity formula of Caffarelli, Jerison and Kening for pairs of functions u _±(x, s) in the strip satisfying

Local regularity of weak solutions of semilinear parabolic systems with critical growth

We show that, under so called controllable growth conditions, any weak solution in the energy class of the semilinear parabolic system