Hessian Equations Research Articles

Hessian equations on closed Hermitian manifolds

In this paper, using the technical tools in \cite{TW5}, we solve the complex Hessian equation on closed Hermitian manifolds, which generalizes the the K\"ahler case results in \cite{HMW} and \cite{DK}.

The convexity of radially symmetric m-subharmonic functions

We consider radially symmetric m-subharmonic functions on the unit ball. We study their convexity and their relation with a solution of the complex Hessian equations. Furthermore, we consider the ordered vector space of radially symmetric m-subharmonic functions which is a Riesz space.

Bifurcation and admissible solutions for the Hessian equation

We study the following eigenvalue problem of k-Hessian equation{Sk(D2u)=λkf(−u)inB,u=0on∂B. Global bifurcation result is established for this problem. As applications of the bifurcation result, we determine the intervals of λ for the existence, nonexistence, uniqueness and multiplicity of radially symmetric k-admissible solutions for this problem.

Asymptotic behaviour of hessian equation with boundary blow up

We consider the boundary blow up problem for k-hessian equation with nonlinearities of power and of exponential type, and prove their existence, uniqueness and asymptotic behaviour. Moreover we also show that their perturbed problem has a unique positive solution, which satisfies some asymptotic behaviors to unperturbed problems under appropriate structure hypotheses for perturbed terms.

Singular solutions to conformal Hessian equations

The authors show that for any e ∈]0, 1[, there exists an analytic outside zero solution to a uniformly elliptic conformal Hessian equation in a ball B ⊂ ℝ5 which belongs to C 1,e (B) \C 1,e +(B).

Hessian measures on m-polar sets and applications to the complex Hessian equations

In this paper, we first study the convergence in m-capacity. These results are used to obtain an extension of a Kolodziej’s result and Ahag et al.’s result to the class .

Liouville and Calabi-Yau type theorems for complex Hessian equations

We prove a Liouville type theorem for entire maximal $m$-subharmonic functions in $\mathbb C^n$ with bounded gradient. This result, coupled with a standard blow-up argument, yields a (non-explicit) a priori gradient estimate for the complex Hessian equation on a compact K\ahler manifold. This terminates the program, initiated by Hou, Ma and Wu, of solving the non-degenerate Hessian equation on such manifolds in full generality. We also obtain, using our previous work, continuous weak solutions in the degenerate case for the right hand side in some $L^p, $ with sharp bound on $p$.

On uniform estimate of complex elliptic equations on closed Hermitian manifolds

In this paper, we study Hessian equations and complex quotient equations on closed Hermitian manifolds. We directly derive the uniform estimate for the admissible solution. As an application, we solve general Hessian equations on closed Kahler manifolds.

A necessary and a sufficient condition for the existence of the positive radial solutions to Hessian equations and systems with weights

In this article, we consider the existence of positive radial solutions for Hessian equations and systems with weights and we give a necessary condition as well as a sufficient condition for a positive radial solution to be large. The method of proving theorems is essentially based on a successive approximation technique. Our results complete and improve a work published recently by Zhang and Zhou (existence of entire positive k-convex radial solutions to Hessian equations and systems with weights. Applied Mathematics Letters, Volume 50, December 2015, Pages 48–55).

A second order estimate for general complex Hessian equations

We derive a priori $C^2$ estimates for the $\chi$-plurisubharmonic solutions of general complex Hessian equations with right-hand side depending on gradients.

Hessian equations on exterior domain

In the paper, we consider the Hessian equation σk(λ(D2u))=f(x) where f is a positive function outside a bounded domain of Rn, n≥3 and f(x)=1+O(|x|−β) for some β>2 at infinity. Using the Perron's method we prove the existence and uniqueness for viscosity solutions of exterior Dirichlet problem with prescribed asymptotic behavior at infinity. There are examples to show that the result is optimal. This is an extension of the theorems given by Bao–Li–Li in [2] for f≡1 and Bao–Li–Zhang in [3] for k=n.

On estimates for the Fu–Yau generalization of a Strominger system

Abstract We study an equation proposed by Fu and Yau as a natural n-dimensional generalization of a Strominger system that they solved in dimension 2. It is a complex Hessian equation with right-hand side depending on gradients. Building on the methods of Fu and Yau, we obtain C 0 {C^{0}} , C 2 {C^{2}} , and C 2 , α {C^{2,\alpha}} a priori estimates. We also identify difficulties in extending the Fu–Yau arguments for non-degeneracy from dimension 2 to higher dimensions.

Oblique boundary value problems for augmented Hessian equations II

In this paper, we continue our investigations into the global theory of oblique boundary value problems for augmented Hessian equations. We construct a global barrier function in terms of an admissible function in a uniform way when the matrix function in the augmented Hessian is only assumed regular. This enables us to derive global second derivative estimates in terms of boundary estimates which are then obtained by strengthening the concavity or monotonicity conditions in our previous work on the strictly regular case. Finally we give some applications to existence theorems which embrace standard Hessian equations as special cases.

Weak solutions of complex Hessian equations on compact Hermitian manifolds

We prove the existence of weak solutions of complex $m$-Hessian equations on compact Hermitian manifolds for the non-negative right-hand side belonging to $L^{p}$, $p>n/m$ ($n$ is the dimension of the manifold). For smooth, positive data the equation has recently been solved by Székelyhidi and Zhang. We also give a stability result for such solutions.

A priori estimates for the obstacle problem of Hessian type equations on Riemannian manifolds

We are concerned with A priori estimates for the obstacle problem of a wide class of fully nonlinear equations on Riemannian manifolds. We use new techniques introduced by Bo Guan and derive new results for A priori second order estimates of its singular perturbation problem under fairly general conditions. By approximation, the existence of a $C^{1,1}$ viscosity solution is proved.

The first initial–boundary value problem for Hessian equations of parabolic type on Riemannian manifolds

In this paper, we are concerned with the first initial–boundary value problem for a class of fully nonlinear parabolic equations on Riemannian manifolds. As usual, the establishment of the a prioriC2 estimates is our main part. Based on these estimates, the existence of classical solutions is proved under conditions which are nearly optimal.

Entire Solution of Hessian Equations in ℂ n $\mathbb {C}^{n}$

We prove the existence and the uniqueness of entire solutions of elliptic Hessian equations in \(\mathbb {C}^{n}\) with rotational invariance in an appropriate weighted Holder spaces.

𝐶^{1,1} regularity for an obstacle problem of Hessian equations on Riemannian manifolds

In this paper, we study an obstacle problem for a class of fully nonlinear equations on Riemannian manifolds. Using some new ideas, the C 1 , 1 C^{1,1} regularity for the greatest viscosity solution is established under essentially optimal structure conditions.

Existence of viscosity solutions with asymptotic behavior of exterior problems for Hessian equations

Existence of viscosity solutions with asymptotic behavior of exterior problems for Hessian equations

An interior estimate for convex solutions and a rigidity theorem

We establish an interior C 2 estimate for k + 1 convex solutions to Dirichlet problems of k -Hessian equations. We also use such estimate to obtain a rigidity theorem for k + 1 convex entire solutions of k -Hessian equations in Euclidean space.