Evolution of nonparametric surfaces with speed depending on curvature II. The mean curvature case

We consider an evolution which starts as a flow of smooth surfaces in nonparametric form propagating in space with normal speed equal to the mean curvature of the current surface. The boundaries of the surfaces are assumed to remain fixed. G. Huisken has shown that if the boundary of the domain over which this flow is considered satisfies the “mean curvature” condition of H. Jenkins and J. Serrin (that is, the boundary of the domain is convex “in the mean”) then the corresponding initial boundary value problem with Dirichlet boundary data and smooth initial data admits a smooth solution for all time. In this paper we consider the case of arbitrary domains with smooth boundaries not necessarily satisfying the condition of Jenkins-Serrin. In this case, even if the flow starts with smooth initial data and homogeneous Dirichlet boundary data, singularities may develop in finite time at the boundary of the domain and the solution will not satisfy the boundary condition. We prove, however, existence of solutions that are smooth inside the domain for all time and become smooth up to the boundary after elapsing of a sufficiently long period of time. From that moment on such solutions assume the boundary values in the classical sense. We also give sufficient conditions that guarantee the existence of classical solutions for all time t ≧ 0. In addition, we establish estimates of the rate at which solutions tend to zero as t → ∞.     

Starshaped Locally Convex Hypersurfaces with Prescribed Curvature and Boundary

In this paper we find strictly locally convex hypersurfaces in \(\mathbb {R}^{n+1}\) with prescribed curvature and boundary. The main result is that if the given data admits a strictly locally convex radial graph as a subsolution, we can find a radial graph realizing the prescribed curvature and boundary. As an application we show that any smooth domain on the boundary of a compact strictly convex body can be deformed to a smooth hypersurface with the same boundary (inside the convex body) and realizing any prescribed curvature function smaller than the curvature of the body.     

Non-parametric radially symmetric mean curvature flow with a free boundary

We study the mean curvature flow of radially symmetric graphs with prescribed contact angle on a fixed, smooth hypersurface in Euclidean space. In this paper we treat two distinct problems. The first problem has a free Neumann boundary only, while the second has two disjoint boundaries, a free Neumann boundary and a fixed Dirichlet height. We separate the two problems and prove that under certain initial conditions we have either long time existence followed by convergence to a minimal surface, or finite maximal time of existence at the end of which the graphs develop a curvature singularity. We also give a rate of convergence for the singularity.     

Convexity estimates for the solutions of a class of semi-linear elliptic equations

In this paper, we are concerned with convexity estimates for solutions of a class of semi-linear elliptic equations involving the Laplacian with power-type nonlinearities. We consider auxiliary curvature functions which attain their minimum values on the boundary and then establish lower bound convexity estimates for the solutions. Then we give two applications of these convexity estimates. We use the deformation method to prove a theorem concerning the strictly power concavity properties of the smooth solutions to these semi-linear elliptic equations. Finally, we give a sharp lower bound estimate of the Gaussian curvature for the solution surface of some specific equation by the curvatures of the domain's boundary.     

An integral kernel for weakly pseudoconvex domains

A new explicit construction of Cauchy–Fantappié kernels is introduced for an arbitrary weakly pseudoconvex domain with smooth boundary. While not holomorphic in the parameter, the new kernel reflects the complex geometry and the Levi form of the boundary. Some estimates are obtained for the corresponding integral operator, which provide evidence that this kernel and related constructions give useful new tools for complex analysis on this general class of domains.     

Nonparametric Mean Curvature Flow with Nearly Vertical Contact Angle Condition

For any bounded strictly convex domain $\Omega$ in $\mathbb{R}^n$ with smooth boundary, we find the prescribed contact angle which is nearly perpendicular such that nonparametric mean curvature flow with contact angle boundary condition converge to ones which move by translation. Subsequently, the existence and uniqueness of smooth solutions to the capillary problem without gravity on strictly convex domain are also discussed.     

A necessary condition of solvability for the capillarity boundary of Monge-Ampere equations in two dimensions

In this paper we consider a class of Monge-Ampere equations with a prescribed contact angle boundary value problem on a bounded strictly convex domain in two dimensions. The purpose is to give a sharp necessary condition of solvability for the above mentioned equations. This is achieved by using the maximum principle and introducing a curvilinear coordinate system for Monge-Ampere equations in two dimensions. An interesting feature of our necessary condition is the need for a certain strong restriction between the curvature of the boundary of domain and the boundary condition, which does not appear in the Dirichlet and Neumann boundary values.

     

On vanishing-curvature extensions of Lorentzian metrics

We study the problem of extending Lorentzian metrics, defined on the boundary of a smooth domain in C, into the interior of the domain in such a way that the curvature vanishes. This amounts to solving a nonlinear partial differential equation for matrices with given boundary values. Generalizing and strengthening a result of Berndtsson, we prove that solutions do exist for certain boundary metrics, assuming the existence of a “subsolution.” The proof is based on the continuity method with a result of Coifman and Semmes concerning positive-definite metrics as the starting point.     

Heat content and horizontal mean curvature on the Heisenberg group

We identify the short time asymptotics of the sub-Riemannian heat content for a smoothly bounded domain in the first Heisenberg group. Our asymptotic formula generalizes prior work by van den Berg–Le Gall and van den Berg–Gilkey to the sub-Riemannian context, and identifies the first few coe?cients in the sub-Riemannian heat content in terms of the horizontal perimeter and the total horizontal mean curvature of the boundary. The proof is probabilistic, and relies on a characterization of the heat content in terms of Brownian motion.     

平均曲率的一个积分等式

<正> 在Соболв空间W_(2,0)~2(Ω)中,假定区域Ω的边界Ω是充分光滑的.本文给出了同Ω的平均曲率有关的一个积分等式,并指出了这个等式的一点应用. 设Ω是R~n中的一个单连通域,边界Ω是及R~n中的n-1维Riemann流形.R~n     

Global strong solution to the three dimensional nonhomogeneous incompressible magnetohydrodynamic equations with density‐dependent viscosity and resistivity

In this paper, we consider an initial boundary value problem for the 3‐dimensional nonhomogeneous incompressible magnetohydrodynamic equations with density‐dependent viscosity and resistivity coefficients over a bounded smooth domain. Global in time unique strong solution is proved to exist when the L² norms of initial vorticity and current density are both suitably small with arbitrary large initial density, and the vacuum of initial density is also allowed. Finally, we revisit the Navier‐Stokes model without electromagnetic effect. We find that this initial boundary problem also admits a unique global strong solution under other conditions. In particular, we prove small kinetic‐energy strong solution exists globally in time, which extends the recent result of Huang and Wang.     

On the lowest eigenvalue of the Hodge Laplacian on compact,negatively curved domains

We study the first eigenvalue of the Laplacian acting on differential forms on a compact Riemannian domain, for the absolute or relative boundary conditions. We prove a series of lower bounds when the domain is starlike or p-convex and the ambient manifold has pinched negative curvature. The bounds are sharp for starlike domains. We then compute the asymptotics of the first eigenvalue of hyperbolic balls of large radius. Finally, we give lower bounds also for Euclidean domains.      

Sufficient conditions for constant mean curvature surfaces to be round

We study under what condition a constant mean curvature surface can be round: i) If the boundary of a compact immersed disk type constant mean curvature surface in consists of lines of curvature and has less than 4 vertices with angle , then the surface is spherical; ii) A compact immersed disk type capillary surface with less than 4 vertices in a domain of bounded by spheres or planes is spherical; iii) The mean curvature vector of a compact embedded capillary hypersurface of with smooth boundary in an unbounded polyhedral domain with unbalanced boundary should point inward; iv) If the kth order () mean curvature of a compact immersed constant mean curvature hypersurface of without boundary is constant, then the hypersurface is a sphere. Received: 3 October 2000 / Published online: 1 February 2002     

The Carleman formula for the Helmholtz equation on the plane

We consider the Cauchy problem for the Helmholtz equation in an arbitrary bounded planar domain with Cauchy data only on part of the boundary of the domain. We derive a Carleman-type formula for a solution to this problem and give a conditional stability estimate.     

带权散度型椭圆算子的低阶特征值估计

本文首先给出紧致带边(边界可以为空集)光滑度量测度空间上带权散度型算子的低阶特征值的一个一般不等式,通过使用这个一般不等式,可以得到光滑度量测度空间中有界连通区域上带权散度型算子的低阶特征值的一些万有不等式.     

Rellich type inequalities in domains of the Euclidean space

For smooth functions supported in a domain of the Euclidean space we investigate two Rellich type inequalities with weights which are powers of the distance function. We prove that for an arbitrary plane domain there exist positive Rellich constants in these inequalities if and only if the boundary of the domain is a uniformly perfect set. Moreover, we obtain explicit estimates of constants in function of geometric domain characteristics. Also, we find sharp constants in these Rellich type inequalities for all non-convex domains of dimension d ≥ 2 provided that the domains satisfy the exterior sphere condition with certain restriction on the radius of spheres.     

On the regularity of solutions of optimal transportation problems

We give a necessary and sufficient condition on the cost function so that the map solution of Monge’s optimal transportation problem is continuous for arbitrary smooth positive data. This condition was first introduced by Ma, Trudinger and Wang [24], [30] for a priori estimates of the corresponding Monge–Ampère equation. It is expressed by a socalled cost-sectional curvature being non-negative. We show that when the cost function is the squared distance of a Riemannian manifold, the cost-sectional curvature yields the sectional curvature. As a consequence, if the manifold does not have non-negative sectional curvature everywhere, the optimal transport map cannot be continuous for arbitrary smooth positive data. The non-negativity of the cost-sectional curvature is shown to be equivalent to the connectedness of the contact set between any cost-convex function (the proper generalization of a convex function) and any of its supporting functions. When the cost-sectional curvature is uniformly positive, we obtain that optimal maps are continuous or Hölder continuous under quite weak assumptions on the data, compared to what is needed in the Euclidean case. This case includes the quadratic cost on the round sphere.     

On maps with given Jacobians involving the heat equation

In this paper we present and analyze two new algorithms to construct a smooth diffeomorphism of a domain with prescribed jacobian function. The first one is free from any restriction on the boundary, while the second one produces a diffeomorphism that coincides with the identity map on the boundary of the domain. Both are based on the solution of an initial value problem for the linear heat equation, and the second also uses solutions of the Stokes system of Fluid Mechanics.     

On maps with given Jacobians involving the heat equation

In this paper we present and analyze two new algorithms to construct a smooth diffeomorphism of a domain with prescribed jacobian function. The first one is free from any restriction on the boundary, while the second one produces a diffeomorphism that coincides with the identity map on the boundary of the domain. Both are based on the solution of an initial value problem for the linear heat equation, and the second also uses solutions of the Stokes system of Fluid Mechanics.