In this paper, we consider the $k$-Hessian equation $S_{k}(D^{2}u)=b(x)f(u)\mbox{ in }Ω,\,u=+\infty \mbox{ on }\partialΩ$, where $Ω$ is a smooth, bounded, strictly convex domain in $\mathbb{R}^{N}$ with $N\geq2$, $b\in \rm C^{\infty}(Ω)$ is positive in $Ω$ and may be singular or vanish on the boundary, $f\in C^{\infty}(0,\infty)\cap C[0, +\infty)$ (or $f\in C^{\infty}(\mathbb{R})$) is positive and increasing on $[0, +\infty)$ (or $\mathbb{R}$) and satisfies the Keller-Osserman type condition. We first supply an upper and lower solution method of classical $k$-convex large solutions to the above equation, and then we studied the optimal global estimate and boundary behavior of large solutions. In particular, we investigate the asymptotic behavior of such solutions when the parameters on $b$ tend to the corresponding critical values and infinity.
In this paper, we study Hessian equations with prescribed contact angle boundary value or oblique derivative boundary value and finally derive the a priori global gradient estimate for the admissible solutions.
In this paper we study the {\it a priori} gradient estimates for admissible solutions to Neumann boundary value problem of fully nonlinear Hessian equations on Riemannian manifolds. We firstly derive an interior gradient estimates for admissible solutions, then we obtain boundary gradient estimates based on the interior gradient estimates we have got.
We study fully nonlinear parabolic equations in nondivergence form with oblique boundary conditions. An optimal and global Calderón-Zygmund estimate is obtained by proving that the Hessian of the viscosity solution to the oblique boundary problem is as integrable as the nonhomogeneous term in $L^{p}$ spaces under minimal regularity requirement on the nonlinear operator, the boundary data and the boundary of the domain.
In this paper, we study global regularity for oblique boundary value problems of augmented Hessian equations for a class of general operators. By assuming a natural convexity condition of the domain together with appropriate convexity conditions on the matrix function in the augmented Hessian, we develop a global theory for classical elliptic solutions by establishing global a priori derivative estimates up to second order. Besides the known applications for Monge-Amp`ere type operators in optimal transportation and geometric optics, the general theory here embraces prescribed mean curvature p…
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 st…
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 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 priori C^2 estimates is our main part. Based on these estimates, the existence of classical solutions is proved under conditions which are nearly optimal.
In bounded domains, without any geometric conditions, we study the existence and uniqueness of globally Lipschitz and interior strong C^{1,1}, (and classical C^2), solutions of general semilinear oblique boundary value problems for degenerate, (and non-degenerate), augmented Hessian equations, with strictly regular associated matrix functions. By establishing local second derivative estimates at the boundary and proving viscosity comparison principles, we show that the solution is correspondingly smooth near boundary points where the appropriate uniform convexity is satisfied.
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 $u$…
We study the asymptotic behavior of C^2-evolutions u = u(x,t) under a given action of the m-Hessian evolution operators and boundary conditions. We obtain sufficient (close to necessary) conditions for the convergence of solutions to the first intial-boundary value problem for the m-Hessian evolution equations to stationary functions as t tends to infinity.
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 a new class of modified-Hessian equations, closely related to the Optimal Transportation Equation, will be introduced and studied. In particular, the existence of globally smooth, classical solutions of these equations satisfying the second boundary value problem will be proven. This proof follows a standard method of continuity argument, which subsequently requires various a priori estimates to be made on classical solutions. These estimates are modifications and generalise the corresponding estimates of Trudinger and Wang for the Optimal Transportation Equation. Of particular n…
In this paper, we derive a priori estimates for the gradient and second order derivatives of solutions to a class of Hessian type fully nonlinear parabolic equations with the first initial-boundary value problem on Riemannian manifolds. These a priori estimates are derived under conditions which are nearly optimal. Especially, there are no geometric restrictions on the boundary of the Riemannian manifolds. And as an application, the existence of smooth solutions to the first initial-boundary value problem even for infinity time is obtained.
In this paper, we consider the Neumann problem of a class of mixed complex Hessian equations, and establish the global C^1 estimates a nd reduce the global second derivative estimate to the estimate of double normal second derivatives on the boundary. In particular, we can prove the global C^2 estimates and the existence theorems when k=n.
Applying ideas of fractional analogue of Monge-Ampére operator by L. Caffarelli and F. Charro, we consider an analogue of fractional k-Hessian operators expressed as concave envelopes of fractional linear operators, and reproduce the same regularity results when k=2. Under the set up of global solutions prescribing data at infinity and global barriers, the key estimate is to prove that fractional 2-Hessian operator is strictly elliptic. Then we can apply nonlocal Evans-Krylov theorem to prove such solutions are classical.
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 global C^2 estimates for a class of mixed Hessian equations with Neumann boundary condition, and obtain the existence theorem of k-admissible solutions for the classical Neumann problem of these mixed Hessian equations.
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 consider a partial differential equation that arises in the coarse-grained description of epitaxial growth processes. This is a parabolic equation whose evolution is governed by the competition between the determinant of the Hessian matrix of the solution and the biharmonic operator. This model might present a gradient flow structure depending on the boundary conditions. We first extend previous results on the existence of stationary solutions to this model for Dirichlet boundary conditions. For the evolution problem we prove local existence of solutions for arbitrary data and global existe…
The author extends previous results to general classes of equations under weaker assumptions obtained in 2016 by Bao, Dong and Jiao concerning the study of the regularity of solutions for the first initial-boundary value problem for parabolic Hessian equations on Riemannian manifolds.
This paper establishes the global curvature estimate for the $n-2$ curvature equation with the general right hand side which partially solves this longstanding problem.
In this paper, we prove some rigidity theorems for the entire 2-convex solutions of 2-Hessian equation in Euclidean space. As an application, we obtain a Bernstein type theorem for global special Lagrangian graphs.
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 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.
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.
In this paper, we establish the second order estimates of solutions to the first initial-boundary value problem for general Hessian type fully nonlinear parabolic equations on Riemannian manifolds. The techniques used in this article can work for a wide range of fully nonlinear PDEs under very general conditions.
In this paper, we obtain some important inequalities of Hessian quotient operators, and global $C^2$ estimates of the Neumann problem of Hessian quotient equations. By the method of continuity, we establish the existence theorem of $k$-admissible solutions of the Neumann problem of Hessian quotient equations.
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 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 prove global and local upper bounds for the Hessian of log positive solutions of the heat equation on a Riemannian manifold. The metric is either fixed or evolves under the Ricci flow. These upper bounds supplement the well-known global lower bound.
In this paper, we proved that any 2-convex solution $u$ of $σ_2(D^2u)=1$ with a quadratic growth must be a quadratic polynomial in $\mathbb{R}^n\ (n\geq 3 )$ by using a Pogorelov estimate and the global gradient estimate. And we give a positive answer to the unresolved issue in \cite{CX}.
Oblique boundary value problems for n-1type augmented Hessian equations
Center for Mathematics and Interdisciplinary Sciences, Fudan University（复旦大学数学与交叉科学研究院）；Shanghai Institute for Mathematics and Interdisciplinary Sciences (SIMIS)（上海数学与交叉学科研究院）；School of Mathematics and Shing-Tung Yau Center of Southeast University, Southeast University（东南大学数学学院及苏步青应用数学中心）
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 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.
In this paper we study the evolution problem \[ \left\lbrace\begin{array}{ll} u_t (x,t)- λ_j(D^2 u(x,t)) = 0, & \text{in } Ω\times (0,+\infty), \\ u(x,t) = g(x,t), & \text{on } \partial Ω\times (0,+\infty), \\ u(x,0) = u_0(x), & \text{in } Ω, \end{array}\right. \] where $Ω$ is a bounded domain in $\mathbb{R}^N$ (that verifies a suitable geometric condition on its boundary) and $λ_j(D^2 u)$ stands for the $j-$st eigenvalue of the Hessian matrix $D^2u$. We assume that $u_0 $ and $g$ are continuous functions with the compatibility condition $u_0(x) = g(x,0)$, $x\in \partial Ω$. We show that the (…
We give a sharp estimate of the modulus of continuity of the solution to the Dirichlet problem for the complex Hessian equation of order $m$ ($1 \leq m \leq n$) with a continuous right hand side and a continuous boundary data in a bounded strongly $m$-pseudoconvex domain $\Om \Subset \C^n$. Moreover when the right hand side is in $L^p(\Om) $, for some $p > n/m$ and the boundary value function is $C^{1,1}$ we prove that the solution is Hölder continuous.
A delicate problem is to obtain existence of solutions to the boundary blow-up elliptic equation% \begin{equation*} σ_{k}^{1/k}\left( λ\left( D^{2}u\right) \right) =g\left( u\right) \text{ in }Ω\text{, }\underset{x\rightarrow x_{0}}{\lim }% u\left( x\right) =+\infty \text{ }\forall x_{0}\in \partial Ω\text{,} \end{equation*}% where $σ_{k}^{1/k}\left( λ\left( D^{2}u\right) \right) $ is the $k$-Hessian operator and $Ω\subset \mathbb{R}^{N}$ is a smooth bounded domain. Our goal is to provide a necessary and sufficient condition on $g$ to ensure existence of at least one positive blow-up solution.…
This work presents the construction of the existence theory of radial solutions to the elliptic equation \begin{equation}\nonumber Δ^2 u = (-1)^k S_k[u] + λf(x), \qquad x \in B_1(0) \subset \mathbb{R}^N, \end{equation} provided either with Dirichlet boundary conditions \begin{eqnarray}\nonumber u = \partial_n u = 0, \qquad x \in \partial B_1(0), \end{eqnarray} or Navier boundary conditions \begin{equation}\nonumber u = Δu = 0, \qquad x \in \partial B_1(0), \end{equation} where the $k-$Hessian $S_k[u]$ is the $k^{\mathrm{th}}$ elementary symmetric polynomial of eigenvalues of the Hessian matrix…
