Hessian Equations Research Articles

Mixed Hessian inequalities and uniqueness in the class $$\mathcal {E}(X,\omega ,m)$$ E ( X , ω , m )

We prove a general inequality for mixed Hessian measures by global arguments. Our method also yields a simplification for the case of complex Monge–Ampere equation. Exploiting this and using Kolodziej’s mass concentration technique we also prove the uniqueness of the solutions to the complex Hessian equation on compact Kahler manifolds in the case of probability measures vanishing on (Formula presented.)-polar sets.

Regularity for an obstacle problem of Hessian equations on Riemannian manifolds

In this paper, we study the regularity for solutions to an obstacle problem of Hessian type fully nonlinear equations on Riemannian manifolds. As an application, the existence of a C1,1 solution is proved.

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

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

Existence of Solutions for Jacobian and Hessian Equations Under Smallness Assumptions

We discuss three types of problems. The first one involves Jacobian equations and the two others involve Hessian equations. We proceed by fixed point, obtaining the results under a smallness assumption.

On the exterior Dirichlet problem for Hessian equations

In this paper, we establish a theorem on the existence of the solutions of the exterior Dirichlet problem for Hessian equations with prescribed asymptotic behavior at infinity. This extends a result of Caffarelli and Li (2003) for the Monge-Ampère equation to Hessian equations.

A priori estimates for complex Hessian equations

We prove some $L^{\infty}$ a priori estimates as well as existence and stability theorems for the weak solutions of the complex Hessian equations in domains of $C^n$ and on compact K\"ahler manifolds. We also show optimal $L^p$ integrability for m-subharmonic functions with compact singularities, thus partially confirming a conjecture of Blocki. Finally we obtain a local regularity result for $W^{2,p}$ solutions of the real and complex Hessian equations under suitable regularity assumptions on the right hand side. In the real case the method of this proof improves a result of Urbas.

Hölder Continuous Solutions to Complex Hessian Equations

We prove the H\"older continuity of the solution to complex Hessian equation with the right hand side in $L^p$, $p>\frac{n}{m}$, $1< m< n$, in a $m$-strongly pseudoconvex domain in $\mathbb{C}^n$ under some additional conditions on the density near the boundary and on the boundary data.

Multi-valued solutions to a class of parabolic Monge-Ampère equations

In this paper, we investigate the multi-valued solutions of a class of parabolic Monge-Ampere equation $-u_{t}\det(D^{2}u)=f$. Using the Perron method, we obtain the existence of finitely valued and infinitely valued solutions to the parabolic Monge-Ampere equations. We generalize the results of elliptic Monge-Ampere equations and Hessian equations.

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.

The Minkowski problem, new constant curvature surfaces in ℝ3, and some applications

Abstract Let 𝔪 ∈ ℕ ${\mathfrak {m}\in \mathbb {N}}$ , 𝔪 ≥ 2 ${\mathfrak {m}\ge 2}$ , and let { p j } j = 1 𝔪 ${\lbrace p_j\rbrace _{j=1}^\mathfrak {m}}$ be a finite subset of 𝕊2 such that 0 → ∈ ℝ 3 ${\vec{0}\in \mathbb {R}^3}$ lies in its positive convex hull. In this paper we make use of the classical Minkowski problem, to show the complete family of smooth convex bodies 𝒦 in ℝ3 whose boundary surface consists of an open surface S with constant Gauss curvature (respectively, constant mean curvature) and 𝔪 planar compact discs D ¯ 1 , ... , D ¯ 𝔪 ${\overline{D}_1,\ldots ,\overline{D}_\mathfrak {m}}$ , such that the Gauss map of S is a homeomorphism onto 𝕊 2 - { p j } j = 1 𝔪 ${\mathbb {S}^2-\lbrace p_j\rbrace _{j=1}^\mathfrak {m}}$ and D j ⊥ p j ${D_j\bot p_j}$ , for all j . ${j.}$ We derive applications to existence of harmonic diffeomorphisms between domains of 𝕊2, existence of capillary surfaces in ℝ3, and a Hessian equation of Monge–Ampère type.

The obstacle problem for Hessian equations on Riemannian manifolds

In this paper, we consider the obstacle problem for a class of fully nonlinear equations on Riemannian manifolds. The C1,1 regularity of the greatest viscosity solutions is established.

The Cauchy problem for indefinite improper affine spheres and their Hessian equation

We give a conformal representation for indefinite improper affine spheres which solve the Cauchy problem for their Hessian equation. As consequences, we can characterize their geodesics and obtain a generalized symmetry principle. Then, we classify the helicoidal indefinite improper affine spheres and find a new family with geodesically complete non-flat affine metric. Moreover, we present interesting examples with singular curves and isolated singularities.

Singularities and Liouville theorems for some special conformal Hessian equations

Singularities and Liouville theorems for some special conformal Hessian equations

Radial symmetric solution of complex Hessian equation in the unit ball

In this note, by using fixed point theorems, we show the existence of a radial symmetric solutions for certain k-complex Hessian equation in the unit ball.

The exterior Dirichlet problem for special Lagrangian equations in dimensions [formula omitted

In this paper, we establish the existence theorem for the exterior Dirichlet problem for special Lagrangian equations with prescribed asymptotic behavior at infinity. This extends the previous results on Monge–Ampère equations and Hessian equations to special Lagrangian equations in dimensions n≤4, which is from calibrated geometry. More generally, we prove that the result is also true for Hessian quotient equations with 0≤l<k≤n in dimensions n≥3.

Singularities of improper affine maps and their Hessian equation

We study improper affine spheres with some admissible singularities, called improper affine maps, and associated to the unimodular Hessian equation. In particular, we characterize when a curve of R3 is the singular curve of some improper affine map with prescribed cuspidal edges and swallowtails. Also, we consider improper affine maps with isolated singularities and show some similarities and differences between the Hessian +1 equation and the Hessian −1 equation. As a consequence, we construct global examples with the desired singularities.

Solutions to degenerate complex Hessian equations

Let (X,ω) be an n-dimensional compact Kähler manifold and fix an integer m such that 1⩽m⩽n. We study degenerate complex Hessian equations of the form (ω+ddcφ)m∧ωn−m=F(x,φ)ωn. 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.

Singular solutions of Hessian elliptic equations in five dimensions

We show that for any δ∈[0,1) there exists a homogeneous order 2−δ analytic outside zero solution to a uniformly elliptic Hessian equation in R5.

Viscosity solutions to complex Hessian equations

We study viscosity solutions to complex Hessian equations. In the local case, we consider Ω a bounded domain in Cn, β the standard Kähler form in Cn and 1⩽m⩽n. Under some suitable conditions on F, g, we prove that the equation (ddcφ)m∧βn−m=F(x,φ)βn, φ=g on ∂Ω admits a unique viscosity solution modulo the existence of subsolution and supersolution. If moreover, the datum is Hölder continuous then so is the solution. In the global case, let (X,ω) be a compact Hermitian homogeneous manifold where ω is an invariant Hermitian metric (not necessarily Kähler). We prove that the equation (ω+ddcφ)m∧ωn−m=F(x,φ)ωn has a unique viscosity solution under some natural conditions on F.

Upper bounds for the eigenvalues of Hessian equations

We prove some upper bounds for the Dirichlet eigenvalues of a class of fully nonlinear elliptic equations, namely the Hessian equations