Multiple solutions to Hessian equations

This paper concerns with the multiple solutions of Hessian equations ?? _k (??(D ² u))?=?f (x, u) in a (k ? 1)-convex domain ${\Omega\subset \mathbb{R}^{n}}$ . Using the methods of degree theory and a priori estimates we prove the existence of two or more solutions to the Hessian equations.     

Multi-valued solutions to fully nonlinear uniformly elliptic equations

In this paper, we discuss the existence and regularity of multi-valued viscosity solutions to fully nonlinear uniformly elliptic equations. We use the Perron method to prove the existence of bounded multi-valued viscosity solutions.     

Existence and convexity of local solutions to degenerate Hessian equations

In this work, we prove the existence of convex solutions to the following k-Hessian equation

$S_k[u]=K(y)g(y,u,Du)$

in the neighborhood of a point $(y_0,u_0,p_0)\in {\mathbb{R}}^n\times \mathbb{R}\times {\mathbb{R}}^n$, where $g\in C^{\infty },g(y_0,u_0,p_0)>0$, $K\in C^{\infty }$ is nonnegative near $y_0$, $K(y_0)=0$ and $\text{Rank}(D_y^{2}K)(y_0)\ge n?k+1$.     

Boundary asymptotical behavior of large solutions to complex Hessian equations

We consider the exact asymptotic behavior of Γ-subharmonic solutions to boundary blow-up problems for the complex Hessian equation in bounded domains Ω.     

Singular solutions of Hessian fully nonlinear elliptic equations

We study Hessian fully nonlinear uniformly elliptic equations and show that the second derivatives of viscosity solutions of those equations (in 12 or more dimensions) can blow up in an interior point of the domain. We prove that the optimal interior regularity of such solutions is no more than C1+?, showing the optimality of the known interior regularity result. The same is proven for Isaacs equations. We prove the existence of non-smooth solutions to fully nonlinear Hessian uniformly elliptic equations in 11 dimensions. We study also the possible singularity of solutions of Hessian equations defined in a neighborhood of a point and prove that a homogeneous order 0<α<1 solution of a Hessian uniformly elliptic equation in a punctured ball should be radial.     

An approximation result for solutions of Hessian equations

We show that W ^2,p weak solutions of the k-Hessian equation F _k (D ² u) = g(x) with k≥ 2 can be approximated by smooth k-convex solutions v _j of similar equations with the right hands sides controlled uniformly in C ^0,1 norm, and so that the quantities are bounded independently of j. This result simplifies the proof of previous interior regularity results for solutions of such equations. It also permits us to extend certain estimates for smooth solutions of degenerate two dimensional Monge–Ampère equations to W ^2,p solutions. Supported by an Australian Research Council Senior Fellowship.     

Some interior regularity results for solutions of Hessian equations

We prove monotonicity formulae related to degenerate k-Hessian equations which yield Morrey type estimates for certain integrands involving the second derivatives of the solution. In the special case k=2 we deduce that weak solutions in , , have locally H?lder continuous gradients. In the nondegenerate case we also show that weak solutions in , , have locally bounded second derivatives. Received February 25, 1999 / Accepted June 11, 1999 / Published online April 6, 2000     

Singular solutions of Hessian elliptic equations in five dimensions

We show that for any δ∈[0,1)$\delta \in [0,1)$ there exists a homogeneous order 2−δ$2-\delta$ analytic outside zero solution to a uniformly elliptic Hessian equation in R⁵${\mathbb{R}}^5$.     

Hessian estimates for convex solutions to quadratic Hessian equation

We derive Hessian estimates for convex solutions to quadratic Hessian equation by compactness argument.     

On uniqueness and existence of viscosity solutions to Hessian equations in exterior domains

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

An interior second derivative bound for solutions of Hessian equations

In previous work we showed that weak solutions in of the k-Hessian equation have locally bounded second derivatives if g is positive and sufficiently smooth and p > kn/2. Here we improve this result to p > k(n-1)/2, which is known to be sharp in the Monge-Ampère case k=n > 2. Received June 21, 1999 / Accepted June 12, 2000 / Published online November 9, 2000     

Existence of viscosity solutions for Hessian equations in exterior domains

The Perron method is used to establish the existence of viscosity solutions to the exterior Dirichlet problems for a class of Hessian type equations with prescribed behavior at infinity.     

Existence of multi-valued solutions with asymptotic behavior of Hessian equations

In this paper, we use the Perron method to prove the existence of multi-valued solutions with asymptotic behavior at infinity of Hessian equations.     

Solutions to degenerate complex Hessian equations

Let (X,ω)$(X,\omega )$ be an n-dimensional compact Kähler manifold and fix an integer m   such that 1?m?n$1?m?n$. We study degenerate complex Hessian equations of the form (ω+dd^cφ)^m∧ω^n−m=F(x,φ)ωⁿ${(\omega +d{d}^c\phi )}^m\wedge {\omega }^{n-m}=F(x,\phi ){\omega }^n$. Under some natural conditions on F, this equation has a unique continuous solution. When X is homogeneous and ω is invariant under the Lie group action, we further show that the solution is Hölder continuous.     

Existence and boundary behavior of solutions of Hessian equations with singular right-hand sides

In this paper, we establish the existence of viscosity solutions of Hessian equations with singular right-hand sides and obtain the asymptotic boundary behavior of solutions. The asymptotic results generalize those for Poisson equations and Monge-Ampère equations, and are more precise than obtained from Hopf lemma.     

Weighted eigenvalue problems for Hessian equations

In this paper we consider weighted eigenvalue problems for fully nonlinear elliptic equations involving Hessian operators. In particular we consider a singular weight, which behaves like a Hardy potential and we prove the existence of weak eigenfunctions.     

Boundary blow-up problems for Hessian equations

This paper is concerned with large solutions of Hessian equations , in a strictly k-convex domain , i.e. solutions such that as . We prove existence, in the viscosity sense, of this kind of solutions assuming that g satisfies some natural growth conditions. Non-existence and uniqueness results are given too. Received: 18 November 1997     

The Sobolev inequality for complex Hessian equations

In this paper, we study the complex Hessian equations by an gradient flow method. We prove a Sobolev inequality for \(k\)-plurisubharmonic functions analogous to that for real Hessian equations (Wang in Indiana Univ Math J 43:25–54, 1994; Lecture Notes in Mathematics, vol. 1977, 2009).