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Jiguang Bao

Publications and source records attributed to Jiguang Bao.

At least 19 recordsLinked to original sources
arXiv2026

Existence of classical solutions to the exterior Dirichlet problem for Hessian quotient equations

This paper studies the exterior Dirichlet problem for Hessian quotient equations with nonconstant right-hand sides. We prove the existence of classical admissible solutions with prescribed asymptotic Hessians and establish convergence of the Hessian at infinity. The main difficulties are obtaining second-order estimates on expanding annuli that are uniform in the outer radius and deriving Hessian convergence under an integral tail condition with no prescribed decay rate. These are resolved through a radius-independent boundary-to-interior estimate and a blow-down argument. We also allow nonradial perturbations of the source. Under stronger pointwise assumptions, we obtain higher-order asymptotic expansions and solutions for every sufficiently large prescribed asymptotic constant.

math.AP↗
arXiv2026

Existence and Nonexistence for Hessian Exterior Dirichlet Problems with \(k\)-Admissible Asymptotic Matrices

We study exterior Dirichlet problems for \(k\)-Hessian equations with prescribed quadratic asymptotics, allowing the asymptotic matrix to be merely \(k\)-admissible and not necessarily positive definite. The key point is that the correct metric at infinity is not determined by the asymptotic matrix itself, but by the coefficient matrix obtained by linearizing the \(k\)-Hessian operator at this matrix. This gives the exterior barriers and subsolutions needed to solve the Dirichlet problem, both in viscosity and smooth settings, for all sufficiently large asymptotic constants. In the case of smooth, strictly star-shaped domains with strictly \((k-1)\)-convex boundary, we complete the characterization of existence and nonexistence through a linearized capacitary comparison and a tangential-trace contradiction on the inner boundary.

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arXiv2026

Optimal Asymptotic Behavior at Infinity for Solutions of the Supercritical Lagrangian Mean Curvature Equation in Exterior Domains

We study the asymptotic behavior at infinity of solutions to the supercritical Lagrangian mean curvature equation \[ \sum_{i=1}^n \arctan λ_i(D^2u)=θ+f(x) \] on exterior domains in \(\mathbb R^n\), \(n\ge 2\), where \(|θ|>((n-2)π)/2\). The perturbation \(f\) is assumed to be locally Lipschitz near infinity and to satisfy a decay condition with rate \(β>0\). The main new ingredient is a scale-dependent difference quotient argument, combined with a nonlocal potential method, which avoids differentiating \(f\) twice and yields quantitative Hessian convergence under only Lipschitz regularity. We establish optimal asymptotic expansions in all dimensions and for all decay rates \(β>0\), including the critical logarithmic cases. This improves previous results requiring higher regularity of \(f\) and faster decay in \cite{BJ2026}.

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arXiv2026

On the second boundary value problem for Lagrangian mean curvature type equation

This article is concerned with the second boundary value problem of the Lagrangian mean curvature type equation arising from special Lagrangian geometry. By the parabolic method, we consider a fully nonlinear parabolic equation with oblique derivative boundary condition, and show the long time existence and convergence of the flow. It follows that the existence and uniqueness of the smooth uniformly convex solution are obtained, which generalizes the Brendle--Warren's theorem about minimal Lagrangian diffeomorphism in Euclidean metric space.

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arXiv2026

The mean curvature type hypersurfaces with prescribed gradient image

In this paper, we consider the existence of mean curvature type hypersurfaces with prescribed gradient image. Let $Ω$ and $\tildeΩ$ be uniformly convex bounded domains in $\mathbb{R}^n$ with smooth boundary. We show that there exists unique convex solutions for the second boundary value problem of mean curvature type equations.

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arXiv2026

Lagrangian Mean Curvature Equations on exterior domains

We introduce an extended exterior $(K,K^{\prime},α_0)$--quasiconformal mapping method to study the asymptotic behavior at infinity of solutions to the supercritical phase Lagrangian mean curvature equation \[ \sum_{i=1}^{n} \arctan λ_i(D^2u) = θ+ f(x) \] on exterior domains in $\mathbb{R}^n$, where the constant $|θ|\in((n-2)π/2,nπ/2)$, $n\geq 2$, and $f=O(|x|^{-β})$ is a perturbation term with the sharp decay condition $β>2$ at infinity. Our work generalizes the classical exterior Bernstein-type theorem for the special Lagrangian equation ($f\equiv0$) established by Li--Li--Yuan [Adv. Math. (2020)]. Via Perron's method, we solve the corresponding Dirichlet problem outside a bounded, uniformly convex domain, prescribing asymptotic behavior at infinity. For $n \geq 3$, we establish existence and uniqueness of viscosity solutions in both the supercritical phase case with $f \not\equiv 0$ and the subcritical phase case with $f \equiv 0$. This extends earlier work by Li [Trans. Amer. Math. Soc. (2019)] on the exterior Dirichlet problem for the special Lagrangian equation ($f \equiv 0$) under weaker regularity assumptions on the interior boundary and boundary data.

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arXiv2026

A Liouville theorem for ancient solutions of the parabolic Monge-Ampère equation with periodic data

This article is concerned with the parabolic Monge-Ampère equation $-u_t\det D_x^2u=f$, where $f=f_1(x)f_2(t)$ and $f_1,f_2$ are positive periodic functions. We prove that any classical parabolically convex ancient solution $u$ must be of the form $-τt+p(x)+v(x,t)$, where $τ$ is a positive constant, $p(x)$ is a convex quadratic polynomial, and $v$ inherits both the spatial and temporal periodicity from $f$. This work extends previous contributions by Caffarelli-Li \cite{cl04} on periodic frameworks for the elliptic Monge-Ampère equations, and generalizes Zhang-Bao \cite{zb18}'s Liouville theorem for $f_2\equiv1$ in parabolic case.

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arXiv2026

Liouville theorem and sharp solvability for solutions of the parabolic Monge-Ampère equation with periodic data

We prove a Liouville Theorem for ancient solutions of the parabolic Monge-Ampère equation with smooth periodic data, generalizing Caffarelli-Li's result \cite{cl04} in 2004 to the parabolic background. To achieve this, we obtain a necessary and sufficient condition for the existence of the smooth periodic solution of the equation $\left(1-u_t\right)\det \left(D_x^2u+I\right)=f$ in $\mathbb{R}^{n+1}$, where $f$ is smooth and periodic in both spatial and temporal variables. This parabolic existence theorem parallels the elliptic counterpart established by Li \cite{l90} in 1990.

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arXiv2025

Optimal asymptotic expansion of entire solutions to Monge-Ampère equation with $C^α$ perturbed periodic data

We consider the asymptotic behavior at infinity of solution $u$ to Monge-Ampère equation $\det(D^2u)=f$ in $\rn$, where $f$ is a perturbation of a periodic function and is only assumed to be Hölder continuous, compared to the previous work that $f$ is at least $C^{1,\az}$. The consequence established in this paper, by a nonlocal method, is that the difference between $u$ and a quadratic polynomial is asymptotically close to a periodic function.

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arXiv2025

Asymptotic behavior at infinity and existence of solutions to the Lagrangian mean curvature flow in $\mathbb R^{n+1}_-$

This paper investigates the asymptotic behavior at infinity of ancient solutions to the Lagrangian mean curvature flow. Under conditions that admit Liouville type rigidity theorems, we prove that every classical solution converges at infinity to the sum of a quadratic polynomial in $x$ and a linear function in $t$, with an explicitly derived exponential rate of convergence. As a critical part of the proof framework of this paper, we establish the existence of a global viscosity solution with prescribed asymptotic behavior at infinity, featuring two key innovations: (i) applicability to all dimensions $n\geq 2$, and (ii) no requirement that the Hessian matrix of the prescribed quadratic term be positive definite or close to a scalar multiple of the identity matrix. These results establish the relationship between Liouville type rigidity, asymptotic analysis at infinity, and the existence of viscosity solutions.

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arXiv2024

Two Necessary and Sufficient Conditions to the Solvability of the Exterior Dirichlet Problem for the Monge-Ampère Equation

The present paper provides two necessary and sufficient conditions for the existence of solutions to the exterior Dirichlet problem of the Monge-Ampère equation with prescribed asymptotic behavior at infinity. By an adapted smooth approximation argument, we prove that the problem is solvable if and only if the boundary value is semi-convex with respect to the inner boundary, which is our first proposed new concept. Along the lines of Perron's method for Laplace equation, we obtain the threshold for solvability in the asymptotic behavior at infinity of the solution, and remove the $C^2$ regularity assumptions on the boundary value and on the inner boundary which are required in the proofs of the corresponding existence theorems in the recent literatures.

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arXiv2023

Existence and asymptotic behavior of entire large solutions for Hessian equations

In this paper, we give some existence and nonexistence results for nonradial entire large solutions of the Hessian equation $S_k\left(D^2 u\right)=b(x) u^γ$ in the sublinear case $0<γ<k$. The exact asymptotic behavior of large solutions at infinity is also studied when $b(x)$ is the oscillation of a radial function $|x|^{-l}$ at infinity for $l\leq k-1$.

math.AP↗
arXiv2023

Interior derivative estimates and Bernstein theorem for Hessian quotient equations

In this paper, we obtain the interior derivative estimates of solutions for elliptic and parabolic Hessian quotient equations. Then we establish the Bernstein theorem for parabolic Hessian quotient equations, that is, any parabolically convex solution $u=u(x,t)\in C^{4,2}(\mathbb{R}^n\times (-\infty,0])$ for $-u_t\frac{S_n(D^2u)}{S_l(D^2u)}=1$ in $\mathbb{R}^n\times (-\infty,0]$ must be the form of $u=-mt+P(x)$ with $m>0$ being a constant and $P$ being a convex quadratic polynomial.

math.AP↗
arXiv2023

Entire solutions to the parabolic Monge--Ampère equation with unbounded nonlinear growth in time

The Liouville type theorem on the parabolic Monge--Ampère equation $-u_t\det D^2u=1$ states that any entire parabolically convex classical solution must be of form $-t+|x|^2/2$ up to a re-scaling and transformation, under additional assumption that partial derivative with respect to time variable $u_t$ is strictly negative and bounded. In this paper, we study the case when $u_t$ is unbounded, prove an existence result of entire parabolically convex smooth solution and investigate the asymptotic behavior near infinity.

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arXiv2023

Existence of entire solutions to the Lagrangian mean curvature equations in supercritical phase

In this paper, we establish the existence and uniqueness theorem of entire solutions to the Lagrangian mean curvature equations with prescribed asymptotic behavior at infinity. The phase functions are assumed to be supercritical and converge to a constant in a certain rate at infinity. The basic idea is to establish uniform estimates for the approximating problems defined on bounded domains and the main ingredient is to construct appropriate subsolutions and supersolutions as barrier functions. We also prove a nonexistence result to show the convergence rate of the phase functions is optimal.

math.AP↗
arXiv2023

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

In this paper, we establish the gradient and Pogorelov estimates for $k$-convex-monotone solutions to parabolic $k$-Hessian equations of the form $-u_tσ_k(λ(D^2u))=ψ(x,t,u)$. We also apply such estimates to obtain a Liouville type result, which states that any $k$-convex-monotone and $C^{4,2}$ solution $u$ to $-u_tσ_k(λ(D^2u))=1$ in $\mathbb{R}^n\times(-\infty,0]$ must be a linear function of $t$ plus a quadratic polynomial of $x$, under some growth assumptions on $u$.

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