In this paper we study the Dirichlet problem for a class of Hessian type equation with its structure as a combination of elementary symmetric functions on Hermitian manifolds. Under some conditions with the initial data on manifolds and admissible subsolutions, we derive a priori estimates for this complex mixed Hessian equation and solvability of the corresponding Dirichlet problem.
In this paper, using the technical tools in \cite{TW5}, we solve the complex Hessian equation on closed Hermitian manifolds, which generalizes the the Kähler case results in \cite{HMW} and \cite{DK}.
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.
In this paper, we derive an \emph{a priori} second order estimate for solutions which are in $Γ_{k+1}$ cone to a class of complex Hessian equations with both sides of the equation depending on the gradient on compact Hermitian manifolds.
We find sharp bounds for the norm inequality on a Pseudo-hermitian manifold, where the L^2 norm of all second derivatives of the function involving horizontal derivatives is controlled by the L^2 norm of the sub-Laplacian. Perturbation allows us to get a-priori bounds for solutions to sub-elliptic PDE in non-divergence form with bounded measurable coefficients. The method of proof is through a Bochner technique. The Heisenberg group is seen to be en extremal manifold for our inequality in the class of manifolds whose Ricci curvature is non-negative.
We solve the classical Dirichlet problem for a general complex Hessian equation on a small ball in $\bC^n$. Then, we show that there is a continuous solution, in pluripotential theory sense, to the Dirichlet problem on compact Hermitian manifolds with boundary that equipped locally conformal Kähler metrics, provided a subsolution.
In this paper, we derive $C^2$ estimates for a class of mixed Hessian type equations with Dirichlet boundary condition, and obtain the existence theorem of admissible solutions for the classical Dirichlet problem of these mixed Hessian type 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 for the Monge-Ampère equation to Hessian equations.
In this paper, we consider the Dirichlet problem for a class of Hessian quotient equations on Riemannian manifolds. Under the assumption of an admissible subsolution, we solve the existence and the uniquness for the Dirichlet problem in a domain without any geometric restrictions on the boundary, based on the a priori estimates for the solutions to the Hessian quotient type equations.
We consider the Dirichlet problem for positively homogeneous, degenerate elliptic, concave (or convex) Hessian equations. Under natural and necessary conditions on the geometry of the domain, with the $C^{1,1}$ boundary data, we establish the interior $C^{1,1}$-regularity of the unique (admissible) solution, which is optimal even if the boundary data is smooth. Both real and complex cases are studied by the unified (Bellman equation) approach.
We are concerned with the Dirichlet problem for a class of Hessian type equations. Applying some new methods we are able to establish the $C^2$ estimates for an approximating problem under essentially optimal structure conditions. Based on these estimates, the existence of classical solutions is proved.
In this paper, we prove the existence of $C^{1,1}$-solution to the Dirichlet problem for degenerate elliptic $k$-Hessian equations $S_{k}[u]=f$ under a condition which is weaker than the condition $f^{1/k}\in C^{1,1}(\barΩ)$.
In this paper, we consider the Dirichlet problem for a new class of augmented Hessian equations. Under sharp assumptions that the matrix function in the augmented Hessian is regular and there exists a smooth subsolution, we establish global second order derivative estimates for the solutions to the Dirichlet problem in bounded domains. The results extend the corresponding results in the previous paper [11] from the Monge-Ampere type equations to the more general Hessian type equations.
We derive second order estimates for $χ$-plurisubharmonic solutions of complex Hessian equations with right hand sides depending on gradients on compact Hermitian manifolds.
In this paper we apply various first and second derivative estimates and barrier constructions from our treatment of oblique boundary value problems for augmented Hessian equations, to the case of Dirichlet boundary conditions. As a result we extend our previous results on the Monge-Ampere and k-Hessian cases to general classes of augmented Hessian equations in Euclidean space
In this paper, we study the Dirichlet problem of Hessian quotient equations in exterior domains. By estimating the eigenvalues of the solution, the necessary and sufficient conditions on existence of radial solutions are obtained. Applying the solutions of ODE, the viscosity subsolutions and supersolutions are constructed and then the existence of viscosity solutions for exterior problems is established by the Perron's method.
In this paper, we consider the exterior Dirichlet problem for Hessian quotient equations with the right hand side $g$, where $g$ is a positive function and $g=1+O(|x|^{-β})$ near infinity, for some $β>2$. Under a prescribed generalized symmetric asymptotic behavior at infinity, we establish an existence and uniqueness theorem for viscosity solutions, by using comparison principles and Perron's method. This extends the previous results for Monge--Ampère equations and Hessian equations.
In this paper, we study the Hessian equation with infinite Dirichlet (blow-up) boundary value conditions. Using radial functions and techniques of ordinary differential inequality, we construct various barrier functions (super-solution and sub-solution). Existence and non-existence theorems are proved by those barriers, maximum principle and theory of viscous solutions. Furthermore, generic boundary blow-up rates for the solutions are derived.
A viscosity approach is introduced for the Dirichlet problem associated to complex Hessian type equations on domains in $\C^n$. The arguments are modelled on the theory of viscosity solutions for real Hessian type equations developed by Trudinger. As consequence we solve the Dirichlet problem for the Hessian quotient and special Lagrangian equations. We also establish basic regularity results for the solutions.
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 Kähler manifolds.
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 functions and using them to analyze …
In this paper, we consider the homogeneous complex k-Hessian equation in an exterior domain $\mathbb{C}^n\setminusΩ$. We prove the existence and uniqueness of the $C^{1,1}$ solution by constructing approximating solutions. The key point for us is to establish the uniform gradient estimate and the second order estimate.
In this paper, we establish second order estimates for a general class of fully nonlinear equations with linear gradient terms on compact almost Hermitian manifolds. As an application, we first prove the existence of solutions for the Monge-Ampère equation with linear gradient terms for $(n-1)$-plurisubharmonic functions, originated from Gaudochon conjecture, in the almost Hermitian setting. Second, we solve the Monge-Ampère equation and Hessian equations with linear gradient terms. Third, we give the $C^{\infty}$ a priori estimates for the deformed Hermitian-Yang-Mills equation with supercrit…
We study the variational structure of the complex $k$-Hessian equation on bounded domain $X\subset \mathbb C^n$ with boundary $M=\partial X$. We prove that the Dirichlet problem $σ_k (\partial \bar{\partial} u) =0$ in $X$, and $u=f$ on $M$ is variational and we give an explicit construction of the associated functional $\mathcal{E}_k(u)$. Moreover we prove $\mathcal{E}_k(u)$ satisfies the Dirichlet principle. In a special case when $k=2$, our constructed functional $\mathcal{E}_2(u)$ involves the Hermitian mean curvature of the boundary, the notion first introduced and studied by X. Wang. Earl…
We treat the exterior Dirichlet problem for a class of fully nonlinear elliptic equations of the form $$f(λ(D^2u))=g(x),$$ with prescribed asymptotic behavior at infinity. The equations of this type had been studied extensively by Caffarelli--Nirenberg--Spruck \cite{Caffarelli1985}, Trudinger \cite{Trudinger1995} 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 $f$ is a concave function. In this paper, based on the Perron's method, we establish an exterior existence and uniquen…
Using the established $d$-concavity of the $k$-Hessian type functions $F_k(R)=\log(S_k(R)),$ whose variables are nonsymmetric matrices, we prove $ C^{2, α}(\overlineΩ) $ estimates for strictly $(δ, \widetildeγ_k) $-admissible solutions to the Dirichlet problem without the well-known regularity condition. A necessary condition for the existence of strictly $δ$-admissible solutions to the equations is given. By the method of continuity, we provide some sufficient conditions for the unique solvability in the class of strictly $(δ,\widetildeγ_k)$-admissible solutions to the Dirichlet problem, prov…
We derive the solvability and regularity of the Dirichlet problem for fully non-linear elliptic equations possibly with degenerate right-hand side on Hermitian manifolds, through establishing a quantitative version of boundary estimate under a subsolution assumption. In addition, we construct the subsolution when the background manifold is a product of a closed Hermitian manifold with a compact Riemann surface with boundary.
In this paper, we mainly establish the existence and uniqueness theorem for solutions of the exterior Dirichlet problem for a class of fully nonlinear second-order elliptic equations related to the eigenvalues of the Hessian, with prescribed generalized symmetric asymptotic behavior at infinity. Moreover, we give some new results for the Hessian equations, Hessian quotient equations and the special Lagrangian equations, which have been studied previously.
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 equations in the almost Hermitian setting.
In this paper, we consider the Dirichlet problem for the homogeneous $k$-Hessian equation with prescribed asymptotic behavior at $0\inΩ$ where $Ω$ is a $(k-1)$-convex bounded domain in the Euclidean space. The prescribed asymptotic behavior at $0$ of the solution is zero if $k>\frac{n}{2}$, it is $\log|x|+O(1)$ if $k=\frac{n}{2}$ and $-|x|^{\frac{2k-n}{n}}+O(1)$ if $k<\frac{n}{2}$. To solve this problem, we consider the Dirichlet problem of the approximating $k$-Hessian equation in $Ω\setminus \overline{B_r(0)}$ with $r$ small. We firstly construct the subsolution of the approximating $k$-Hess…
We solve the Dirichlet problem for Monge-Ampère equation for $(n-1)$-PSH functions possibly with degenerate right-hand side, through deriving a quantitative version of boundary estimate under the assumption of $(n-1)$-PSH subsolutions. In addition, we confirm the subsolution assumption on a product of a closed balanced manifold with a compact Riemann surface with boundary.
This note aims to investigate the regularity of a solution to the Dirichlet problem for the complex Hessian equation, which has a density of the $m$-Hessian measure that belongs to $L^q$, for $q\leq\frac nm$.
We study weak geodesics in the space of potentials for the deformed Hermitian-Yang-Mills equation. The geodesic equation can be formulated as a degenerate elliptic equation, allowing us to employ nonlinear Dirichlet duality theory, as developed by Harvey-Lawson. By exploiting the convexity of the level sets of the Lagrangian angle operator in the highest branch, we are able to construct continuous solutions of the associated Dirichlet problem.
In this paper we prove the existence of nonsmooth viscosity solutions for Dirichlet problems involving elementary symmetric functions of the eigenvalues of the complex Hessian.
We consider the Dirichlet problem for two types of degenerate elliptic Hessian equations . New results about solvability of the equations in the $C^{1,1}$ space are provided.
Let $Ω\subseteq M$ be a bounded domain with a smooth boundary $\partialΩ$, where $(M,J,g)$ is a compact, almost Hermitian manifold. The main result of this paper is to consider the Dirichlet problem for a complex Monge-Ampère equation on $Ω$. Under the existence of a $C^{2}$-smooth strictly $J$-plurisubharmonic ($J$-psh for short) subsolution, we can solve this Dirichlet problem. Our method is based on the properties of subsolutions which have been widely used for fully nonlinear elliptic equations over Hermitian manifolds.
