Positive solutions of the Dirichlet problem for the prescribed mean curvature equation

We discuss existence and multiplicity of positive solutions of the prescribed mean curvature problem

     

On the exterior Dirichlet problem for Hessian quotient equations

In this paper, we establish the existence and uniqueness theorem for solutions of the exterior Dirichlet problem for Hessian quotient equations with prescribed asymptotic behavior at infinity. This extends the previous related results on the Monge–Ampère equations and on the Hessian equations, and rearranges them in a systematic way. Based on the Perron's method, the main ingredient of this paper is to construct some appropriate subsolutions of the Hessian quotient equation, which is realized by introducing some new quantities about the elementary symmetric polynomials and using them to analyze the corresponding ordinary differential equation related to the generalized radially symmetric subsolutions of the original equation.     

A priori estimates and existence of solutions to the prescribed centroaffine curvature problem

In this paper we study the prescribed centroaffine curvature problem in the Euclidean space ${\mathbb{R}}^{n+1}$. This problem is equivalent to solving a Monge–Ampère equation on the unit sphere. It corresponds to the critical case of the Blaschke–Santaló inequality. By approximation from the subcritical case, and using an obstruction condition and a blow-up analysis, we obtain sufficient conditions for the a priori estimates, and the existence of solutions up to a Lagrange multiplier.     

The Dirichlet problem for Hessian quotient equations in exterior domains

In this paper, we obtain the uniqueness and existence of viscosity solutions with prescribed asymptotic behavior at infinity to Hessian quotient equations in exterior domains.     

On the Dirichlet problem for the prescribed mean curvature equation over general domains

We study and solve the Dirichlet problem for graphs of prescribed mean curvature in Rn+1 over general domains Ω without requiring a mean convexity assumption. By using pieces of nodoids as barriers we first give sufficient conditions for the solvability in case of zero boundary values. Applying a result by Schulz and Williams we can then also solve the Dirichlet problem for boundary values satisfying a Lipschitz condition.     

The Dirichlet problem for the equation of prescribed scalar curvature in Minkowski space

We prove a maximum principle for the curvature of spacelike admissible solutions of the equation of prescribed scalar curvature in Minkowski space. This enables us to extend to higher dimensions a recent existence result of Bayard for the Dirichlet problem in three and four dimensions. We also prove an interior curvature bound which permits us to prove the existence of locally smooth solutions in the case of spacelike affine boundary data. Uniform convexity of the boundary data is assumed throughout.Received: 11 November 2002, Accepted: 23 January 2003, Published online: 16 May 2003Mathematics Subject Classification (1991): 35J60, 35J65, 53C50John Urbas: Supported by an Australian Research CouncilSenior Research Fellowship     

Infinitely many solutions for the prescribed curvature problem of polyharmonic operator

We consider the following prescribed curvature problem for polyharmonic operator: $$\left\{\begin{array}{llll} D_{m} u = \tilde{K}(y)|u|^{m^*-2}u\; {\rm in}\; \mathbb{S}^N\\ u \quad\; >0\qquad\quad\quad\quad\quad{\rm on}\; \mathbb{S}^N\\ u \quad\; \in H^{m}(\mathbb{S}^N), \end{array} \right.$$ where ${m^*=\frac{2N}{N-2m}, N\geq 2m+1,m \in \mathbb{N}_{+}, \tilde{K}}$ is positive and rationally symmetric, ${\mathbb{S}^N}$ is the unit sphere with the induced Riemannian metric ${g=g_{\mathbb{S}^N},}$ and D _m is the elliptic differential operator of 2m order given by $$\begin{array}{lll}D_m={\prod\limits_{k=1}^m}{\left(-\Delta_g+\frac{1}{4}(N-2k)(N+2k-2)\right)}\end{array}$$ where Δ _g is the Laplace-Beltrami operator on ${\mathbb{S}^N}$ . We will show that problem (P) has infinitely many non-radial positive solutions, whose energy can be arbitrary large.     

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

A priori estimate for the second-order derivatives of solutions to the Dirichlet problem for fully nonlinear parabolic equations

The first initial-boundary-value problem for a uniformly parabolic and uniformly nondegenerate operator is considered. An a priori estimate for an admissible solution is established. In view of the generalized Hessian, it is possible to avoid the growth conditions which are usual in the theory of uniformly elliptic and uniformly parabolic operators. Bibliography: 19 titles. Translated fromProblemy Matematicheskogo Analiza, No. 16. 1997, pp. 112–133.     

Translating solutions to the second boundary value problem for curvature flows

 We consider the flow of strictly convex hypersurfaces driven by curvature functions subject to the second boundary condition and show that they converge to translating solutions. We also discuss translating solutions for Hessian equations. Received: 10 May 2001 / Revised version: 30 January 2002