Hessian Quotient Equations Research Articles

On the Exterior Problem for Parabolic Hessian Quotient Equations

On the Exterior Problem for Parabolic Hessian Quotient Equations

A class of Hessian quotient equations in de Sitter space

Abstract In this paper, we consider the closed spacelike solution to a class of Hessian quotient equations in de Sitter space. Under mild assumptions, we obtain an existence result using standard degree theory based on a priori estimates.

Solvability of Hessian quotient equations in exterior domains

Abstract In this paper, we study the Dirichlet problem of Hessian quotient equations of the form $S_k(D^2u)/S_l(D^2u)=g(x)$ in exterior domains. For $g\equiv \mbox {const.}$ , we obtain the necessary and sufficient conditions on the existence of radially symmetric solutions. For g being a perturbation of a generalized symmetric function at infinity, we obtain the existence of viscosity solutions by Perron’s method. The key technique we develop is the construction of sub- and supersolutions to deal with the non-constant right-hand side g.

On a class of Hessian quotient type equations on Riemannian manifolds

Jiao-Liu [Proc. Amer. Math. Soc. 151 (2023), pp. 569–581] proved the existence and the uniqueness of solutions for a class of k-Hessian equations on Riemannian manifolds using the continuity method. In this article, we use the results of Chen et al. [Calc. Var. Partial Differential Equations 62 (2023), p. 15] to extend their results to the corresponding Hessian quotient equations on Riemannian manifolds.

Serrin-Type Overdetermined Problems for Hessian Quotient Equations and Hessian Quotient Curvature Equations

Serrin-Type Overdetermined Problems for Hessian Quotient Equations and Hessian Quotient Curvature Equations

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.

The Lp Minkowski type problem for a class of mixed Hessian quotient equations

Existence and uniqueness of the admissible solutions to the Lp Minkowski type problem for mixed Hessian operators are proved. By using the new phenomenon of “inverse convexity” of these operators, we show the existence of the geometric convex solutions by the Full Rank Theorem.

The Dirichlet Problem for a Class of Hessian Quotient Equations on Riemannian Manifolds

Abstract In this paper, we consider the Dirichlet problem for a class of Hessian quotient equations involving a gradient term on the right-hand sides on Riemannian manifolds. Under the assumption of an admissible subsolution, we solve the existence and the uniqueness for the Dirichlet problem on compact Riemannian manifold, based on the a priori estimates for the solutions to the Hessian quotient type equations. Compared with the classical results for Hessian type equations, our results do not depend on the convexity assumption for the right-hand side of the equation.

Pogorelov type estimates for a class of Hessian quotient equations in Lorentz-Minkowski space [formula omitted

Let Ω be a bounded domain (with smooth boundary) on the hyperbolic plane Hn(1), of center at origin and radius 1, in the (n+1)-dimensional Lorentz-Minkowski space R1n+1. In this paper, by using a priori estimates, we can establish Pogorelov type estimates of k-convex solutions to a class of Hessian quotient equations defined over Ω⊂Hn(1) and with the vanishing Dirichlet boundary condition.

The complex Hessian quotient flow on compact Hermitian manifolds

<abstract><p>In this paper, we consider the parabolic Hessian quotient equation on compact Hermitian manifolds. By setting up a priori estimates of the admissible solutions, we prove the long-time existence of the solution to the parabolic Hessian quotient equation and its convergence. As an application, we show the solvability of a class of complex Hessian quotient equations, which generalizes the relevant results.</p></abstract>

The Neumann Problem for Parabolic Hessian Quotient Equations

In this paper, we consider the Neumann problem for parabolic Hessian quotient equations. We show that the k-admissible solution of the parabolic Hessian quotient equation exists for all time and converges to the smooth solution of elliptic Hessian quotient equations. Also solutions of the classical Neumann problem converge to a translating solution.

The Neumann problem for degenerate Hessian quotient equations

In this paper, we prove the existence of the [Formula: see text]-solution to the classical Neumann problem for the degenerate elliptic Hessian quotient equation [Formula: see text] under the condition that [Formula: see text].

Pogorelov type estimates for a class of Hessian quotient equations

Pogorelov type estimates of solutions to Hessian equations have been established by Chou-Wang and Li-Ren-Wang. However, as far as we know, Pogorelov type estimates for Hessian quotient equations are still unknown. In this paper, we obtain such estimates for a class of Hessian quotient equations which naturally appears in complex geometry, conformal geometry and differential geometry.

On the exterior Dirichlet problem for a class of fully nonlinear elliptic equations

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 its applications to the Hessian equations, Hessian quotient equations and the special Lagrangian equations, which have been studied previously.

A class of Hessian quotient equations in Euclidean space

In this paper, we consider a class of Hessian quotient equations in Euclidean space. Under some sufficient conditions, we obtain an existence result using standard degree theory based on a priori estimates for solutions to Hessian quotient equations.

The Neumann problem of Hessian quotient equations

In this paper, we obtain some important inequalities of Hessian quotient operators, and global [Formula: see text] estimates of the Neumann problem of Hessian quotient equations. By the method of continuity, we establish the existence theorem of [Formula: see text]-admissible solutions of the Neumann problem of Hessian quotient equations.

The Neumann Problem of Complex Hessian Quotient Equations

Abstract In this paper, we consider the Neumann problem of complex Hessian quotient equations $\frac{\sigma _k (\partial \bar{\partial } u)}{\sigma _l (\partial \bar{\partial } u)} = f(z)$ with $0 \leq l < k \leq n$ and establish the global $C^1$ estimates and reduce the global 2nd derivative estimate to the estimate of double normal 2nd derivatives on the boundary. In particular, we can prove the global $C^2$ estimates and the existence theorem for the Neumann problem of complex Hessian quotient equations $\frac{\sigma _n (\partial \bar{\partial } u)}{\sigma _l (\partial \bar{\partial } u)} = f(z)$ with $0 \leq l < n$ by the method of continuity.

Fully non-linear elliptic equations on compact Hermitian manifolds

We derive a priori estimates for solutions of a general class of fully non-linear equations on compact Hermitian manifolds. Our method is based on ideas that have been used for different specific equations, such as the complex Monge–Ampere, Hessian and inverse Hessian equations. As an application we solve a class of Hessian quotient equations on Kahler manifolds assuming the existence of a suitable subsolution. The method also applies to analogous equations on compact Riemannian manifolds.

Oblique boundary value problems for augmented Hessian equations I

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–Ampère type operators in optimal transportation and geometric optics, the general theory here embraces Neumann problems arising from prescribed mean curvature problems in conformal geometry as well as general oblique boundary value problems for augmented k-Hessian, Hessian quotient equations and certain degenerate equations.

On the exterior Dirichlet problem for Hessian quotient 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 polynomials and using them to analyze the corresponding ordinary differential equation related to the generalized radially symmetric subsolutions of the original equation.