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.
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 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.
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 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.
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…
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.
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 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, 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 $k$-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.
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 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.
We consider the problem of existence and uniqueness of strong solutions $u: Ω\subset \mathbb{R}^n \longrightarrow \mathbb{R}^N$ in $(H^{2}\cap H^{1}_0)(Ω)^N$ to the problem \[\label{1} \tag{1} \left\{ \begin{array}{l} F(\cdot,D^2u ) \,=\, f, \ \ \text{ in }Ω,\\ \hspace{31pt} u\,=\, 0, \ \ \text{ on }\partial Ω, \end{array} \right. \] when $ f\in L^2(Ω)^N$, $F$ is a Carathéodory map and $Ω$ is convex. \eqref{1} has been considered by several authors, firstly by Campanato and under Campanato's ellipticity condition. By employing a new weaker notion of ellipticity introduced in recent work of the…
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 consider the Dirichlet problem for the biharmonic equation on an arbitrary convex domain and prove that the second derivatives of the variational solution are bounded in all dimensions.
We mainly study Pogorelov type $C^2$ estimates for solutions to the Dirichlet problem of Sum Hessian equations. We establish respectively Pogorelov type $C^2$ 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.
The $C^{1,1}$ estimate of the Dirichlet problem for degenerate $k$-Hessian equations with non-homogenous boundary conditions is an open problem, if the right hand side function $f$ is only assumed to satisfy $f^{1/(k-1)} \in C^{1,1}$. In this paper, we solve this problem for convex solutions defined in the strictly convex bounded domain.
Existence of Admissible Subsolutions to the Dirichlet Problem for Symmetric Augmented $k$-Hessian Type Equations in Bounded Domains
Hanoi Pedagogical University 2（河内师范大學二校）；University of Economics - Technology for Industries（經濟與工業技術大學）；Hanoi University of Science and Technology（河內科技大學）；Quang Binh University（廣平大學）
It is well-known that a celebrated Jörgens-Calabi-Pogorelov theorem for Monge-Ampère equations states that any classical (viscosity) convex solution of $\det(D^2u)=1$ in $\mathbb{R}^n$ must be a quadratic polynomial. Therefore, it is an interesting topic to study the existence and uniqueness theorem of such fully nonlinear partial differential equations' Dirichlet problems on exterior domains with suitable asymptotic conditions at infinity. As a continuation of the works of Caffarelli-Li for Monge-Ampère equation and of Bao-Li-Li for $k$-Hessian equations, this paper is devoted to the solvabil…
In this paper, we establish the existence and uniqueness theorem for entire solutions of Hessian equations with prescribed asymptotic behavior at infinity. This extends the previous results on Monge-Ampère equations. Our approach also makes the prescribed asymptotic order optimal within the range preset in exterior Dirichlet problems. In addition, we show a Liouville type result for $k$-convex solutions. This partly removes the $(k+1)$- or $n$-convexity restriction imposed in existing work.
The $k$-Hessian operator $σ_k$ is the $k$-th elementary symmetric function of the eigenvalues of the Hessian. It is known that the $k$-Hessian equation $σ_k(D^2u)=f$ with Dirichlet boundary condition $u=0$ is variational; indeed, this problem can be studied by means of the $k$-Hessian energy $-\int uσ_k(D^2u)$. We construct a natural boundary functional which, when added to the $k$-Hessian energy, yields as its critical points solutions of $k$-Hessian equations with general non-vanishing boundary data. As a consequence, we prove a sharp Sobolev trace inequality for $k$-admissible functions $u$…
In this paper, we show the existence and uniqueness of viscosity solution to the Cauchy-Dirichlet problem for a class of fully nonlinear parabolic equations. This extends recent results of Eyssidieux-Guedj-Zeriahi.
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 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.
In this paper, we prove second derivative estimates together with classical solvability for the Dirichlet problem of certain Monge-Ampere type equations under sharp hypotheses. In particular we assume that the matrix function in the augmented Hessian is regular in the sense used by Trudinger and Wang in their study of global regularity in optimal transportation [28] as well as the existence of a smooth subsolution. The latter hypothesis replaces a barrier condition also used in their work. The applications to optimal transportation and prescribed Jacobian equations are also indicated.
The Dirichlet problem and Dirichlet to Neumann map are analyzed for elliptic equations on a large collection of infinite quantum graphs. For a dense set of continuous functions on the graph boundary, the Dirichlet to Neumann map has values in the Radon measures on the graph boundary.
For a sufficiently regular open bounded set $D \subset R^2$ let us consider the equation $(-Δ)^{1/2} φ(x) = 1$, $x \in D$ with the Dirichlet exterior condition $φ(x) = 0$, $x \in D^c$. $φ$ is the expected value of the first exit time from $D$ of the Cauchy process in $R^2$. We prove that if $D \subset R^2$ is a convex bounded domain then $φ$ is concave on $D$. To show it we study the Hessian matrix of the harmonic extension of $φ$. The key idea of the proof is based on a deep result of Hans Lewy concerning determinants of Hessian matrices of harmonic functions.
In this note, we prove some non-existence results for Dirichlet problems of complex Hessian equations. The non-existence results are proved using the Pohozaev method. We also prove existence results for radially symmetric solutions. The main difference of the complex case with the real case is that we don't know if a priori radially symmetric property holds in the complex case.
