Hessian Equations Research Articles

Complex Hessian-type equations in the weighted m -subharmonic class

UDC 517.5 We study the existence of a solution to a general type of complex Hessian equation on some Cegrell classes. For a given measure μ defined on an m -hyperconvex domain Ω ⊂ ℂ n , under suitable conditions, we prove that the equation χ ( . ) H m ( . ) = μ has a solution that belongs to the class ℰ m , χ ( Ω ) .

On the convexity of general inverse σk equations

We prove that if a level set of a degree n general inverse σk equation f(λ1,⋯,λn):=λ1⋯λn−∑k=0n−1ckσk(λ)=0 is contained in q+Γn for some q∈Rn, where ck are real numbers not necessary to be non-negative and Γn is the positive orthant, then this level set is convex. As an application, this result justifies the convexity of the level set of all general inverse σk type equations, for example, the Monge–Ampère equation, the Hessian equation, the J-equation, the deformed Hermitian–Yang–Mills equation, the special Lagrangian equation, etc. Moreover, we find a numerical condition to verify whether a level set of a general inverse σk equation is contained in q+Γn for some q∈Rn, which is a way to determine the convexity of this level set.

The existence of infinitely many boundary blow-up solutions to the p-k-Hessian equation

Abstract The primary objective of this article is to analyze the existence of infinitely many radial p p - k k -convex solutions to the boundary blow-up p p - k k -Hessian problem σ k ( λ ( D i ( ∣ D u ∣ p − 2 D j u ) ) ) = H ( ∣ x ∣ ) f ( u ) in Ω , u = + ∞ on ∂ Ω . {\sigma }_{k}\left(\lambda \left({D}_{i}\left({| Du| }^{p-2}{D}_{j}u)))=H\left(| x| )f\left(u)\hspace{0.33em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega ,\hspace{0.33em}u=+\infty \hspace{0.33em}\hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.33em}\partial \Omega . Here, k ∈ { 1 , 2 , … , N } k\in \left\{1,2,\ldots ,N\right\} , σ k ( λ ) {\sigma }_{k}\left(\lambda ) is the k k -Hessian operator, and Ω \Omega is a ball in R N ( N ≥ 2 ) {{\mathbb{R}}}^{N}\hspace{0.33em}\left(N\ge 2) . Our methods are mainly based on the sub- and super-solutions method.

Existence and asymptotic behavior of strictly convex solutions for singular k-Hessian equations with nonlinear gradient terms

Abstract In this paper, we mainly consider the singular k-Hessian equations S k ⁢ ( λ ⁢ ( D 2 ⁢ u ) ) = h ⁢ ( x ) ⁢ f ⁢ ( - u ) + g ⁢ ( | D ⁢ u | ) in ⁢ Ω S_{k}(\lambda(D^{2}u))=h(x)f(-u)+g(|Du|)\quad\text{in }\Omega and S k ⁢ ( λ ⁢ ( D 2 ⁢ u ) ) = h ⁢ ( x ) ⁢ f ⁢ ( - u ) ⁢ ( 1 + g ⁢ ( | D ⁢ u | ) ) in ⁢ Ω S_{k}(\lambda(D^{2}u))=h(x)f(-u)(1+g(|Du|))\quad\text{in }\Omega with the Dirichlet boundary condition u = 0 {u=0} on ∂ ⁡ Ω {\partial\Omega} , where Ω ⊂ ℝ N {\Omega\subset\mathbb{R}^{N}} ( N ≥ 2 {N\geq 2} ) is a strictly convex, bounded smooth domain. Using the method of upper and lower solutions and the Karamata regular variation theory, we get new criteria of the existence and asymptotic behavior of strictly convex solutions under different conditions imposed on h, f and g. This problem is more difficult to solve than the k-Hessian problem without gradient terms, and requires additional new conditions in the proof process.

The multiplicity of radial k-convex solutions for an augmented Hessian equation

In this article, we apply the sub-super-solution method to investigate the multiplicity of radial k-convex solutions to an augmented Hessian problem{Sk(λ(D2u−μI))=b(|x|)g(u),x∈Ω,u=+∞,x∈∂Ω, where Ω⊆RN(N≥2) is a ball, k∈1,2,⋯,N, and μ is a positive constant.

Necessary and sufficient conditions for the existence of entire subsolutions to [formula omitted]-Hessian equations

In the paper, we intend to analyze the existence of entire subsolutions to the p-k-Hessian equation Sk1k(λ((Di(|Du|p−2Dju))+α|Du|β(p−1)I))=f(u),x∈Rn,where β = 0 or 1, p≥2 and α≥0 is a constant. The necessary and sufficient conditions on f for the existence of entire subsolutions are established. Since β = 0 or 1 and α≥0 not α≡0, it is very difficult to judge the sign of second order derivative. We will overcome the difficulty by introducing a new technique.

Continuous Solutions for Degenerate Complex Hessian Equation

Continuous Solutions for Degenerate Complex Hessian Equation

Fully non-linear elliptic equations on compact almost Hermitian manifolds

In this paper, we establish a priori estimates for solutions of a general class of fully non-linear equations on compact almost Hermitian manifolds. As an application, we solve the complex Hessian equation and the Monge–Ampère equation for $$(n-1)$$ -plurisubharmonic functions in the almost Hermitian setting.

On a class of obstacle problem for Hessian equations on Riemannian manifolds

In this paper, we establish the a prioriC^{2} estimates for solutions of a class of obstacle problem for Hessian equations on Riemannian manifolds. Some applications are also discussed. The main contribution of this paper is the boundary estimates for second-order derivatives.

On the exterior Dirichlet problem for Hessian-type fully nonlinear elliptic equations

We treat the exterior Dirichlet problem for a class of fully nonlinear elliptic equations of the form [Formula: see text] with prescribed asymptotic behavior at infinity. The equations of this type had been studied extensively by Caffarelli et al. [The Dirichlet problem for nonlinear second-order elliptic equations. III. Functions of the eigenvalues of the Hessian, Acta Math. 155 (1985) 261–301], Trudinger [On the Dirichlet problem for Hessian equations, Acta Math. 175 (1995) 151–164] and many others, and there had been significant discussions on the solvability of the classical Dirichlet problem via the continuity method, under the assumption that [Formula: see text] is a concave function. In this paper, based on Perron’s method, we establish an exterior existence and uniqueness result for viscosity solutions of the equations, by assuming [Formula: see text] to satisfy certain structure conditions as in [L. Caffarelli, L. Nirenberg and J. Spruck, The Dirichlet problem for nonlinear second-order elliptic equations. III. Functions of the eigenvalues of the Hessian, Acta Math. 155 (1985) 261–301; N. S. Trudinger, On the Dirichlet problem for Hessian equations, Acta Math. 175 (1995) 151–164] but without requiring the concavity of [Formula: see text]. The equations in our setting may embrace the well-known Monge–Ampère equations, Hessian equations and Hessian quotient equations as special cases.

Fully nonlinear elliptic equations with gradient terms on compact almost Hermitian manifolds

We establish second order estimates for a general class of fully nonlinear elliptic equations with gradient terms on almost Hermitian manifolds including the deformed Hermitian-Yang-Mills equation and the equation in the proof of Gauduchon conjecture by Székelyhidi-Tosatti-Weinkove. As applications, we also consider the existence of Monge-Ampère equation and Hessian equations.

Interior estimates of derivatives and a Liouville type theorem for parabolic $ k $-Hessian equations

Interior estimates of derivatives and a Liouville type theorem for parabolic $ k $-Hessian equations

The Solutions with Prescribed Asymptotic Behavior for the Exterior Dirichlet Problem of Hessian Equations

The Solutions with Prescribed Asymptotic Behavior for the Exterior Dirichlet Problem of Hessian Equations

Entire subsolutions of a kind of $ \boldsymbol{k} $-Hessian type equations with gradient terms

In this paper, we consider a $ k $-Hessian type equations $ S_k^{\frac{1}{k}}[D^2u+\mu|D u|I] = f(u) $ in $ \mathbb{R}^n $, and provide a necessary and sufficient condition for the solvability of entire admissible subsolutions, which can be regarded as a generalized Keller-Osserman condition. The existence and nonexistence results are proved in different ranges of the parameter $ \mu $ respectively, which embrace the standard Hessian equation case ($ \mu = 0 $) by Ji and Bao (Proc. Amer. Math. Soc. 138: 175–188, 2010) as a typical example. The difference between the semilinear case ($ k = 1 $) and the fully nonlinear case ($ k\ge 2 $) is also concerned.

The Dirichlet problem for the k-Hessian equation on a complex manifold

We solve the Dirichlet problem for $k$-Hessian equations on compact complex manifolds with boundary, given the existence of a subsolution. Our method is based on a second order a priori estimate of the solution on the boundary with a particular gradient scale. The scale allows us to apply a blow-up argument to obtain control on all necessary norms of the solution.

Symmetry of solutions of minimal gradient graph equations on punctured space

In this paper, we study symmetry and existence of solutions of minimal gradient graph equations on punctured space ℝn \ {0}, which include the Monge—Ampère equation, inverse harmonic Hessian equation and the special Lagrangian equation. This extends the classification results on Monge—Ampère equations. Under some conditions, we also give a necessary and sufficient condition for the solvability of the exterior Dirichlet problem in terms of their asymptotic behaviors.

Stability estimates for the complex Monge-Ampère and Hessian equations

A new proof for stability estimates for the complex Monge-Ampère and Hessian equations is given, which does not require pluripotential theory. A major advantage is that the resulting stability estimates are then uniform under general degenerations of the background metric in the case of the Monge-Ampère equation, and under degenerations to a big class in the case of Hessian equations.

Hessian equations of Krylov type on compact Hermitian manifolds

Abstract In this article, we are concerned with the equations of Krylov type on compact Hermitian manifolds, which are in the form of the linear combinations of the elementary symmetric functions of a Hermitian matrix. Under the assumption of the 𝒞-subsolution, we obtain a priori estimates in Γ k − 1 {\Gamma }_{k-1} cone. By using the method of continuity, we prove an existence theorem, which generalizes the relevant results. As an application, we give an alternative way to solve the deformed Hermitian Yang-Mills equation on compact Kähler threefold.

Pogorelov type C2 estimates for Sum Hessian equations and a rigidity theorem

We mainly study Pogorelov type C2 estimates for solutions to the Dirichlet problem of Sum Hessian equations. We establish respectively Pogorelov type C2 estimates for k-convex solutions and admissible solutions under some conditions. Furthermore, we apply such estimates to obtain a rigidity theorem for k-convex solutions of Sum Hessian equations in Euclidean space.

Regularity of degenerate k-Hessian equations on closed Hermitian manifolds

Abstract In this article, we are concerned with the existence of weak C 1 , 1 {C}^{1,1} solution of the k k -Hessian equation on a closed Hermitian manifold under the optimal assumption of the function in the right-hand side of the equation. The key points are to show the weak C 1 , 1 {C}^{1,1} estimates. We prove a Cherrier-type inequality to obtain the C 0 {C}^{0} estimate, and the complex Hessian estimate is proved by using an auxiliary function, which was motivated by Hou et al. and Tosatti and Weinkove. Our result generalizes the Kähler case proved by Dinew et al.