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Jiguang Bao

Publications and source records attributed to Jiguang Bao.

At least 37 records · Page 2Linked to original sources
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 $u$.

math.AP↗
arXiv2022

The exterior Dirichlet problems of Hessian quotient equations

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.

math.AP↗
arXiv2022

Entire solutions of the generalized Hessian inequality

In this paper, we discuss the more general Hessian inequality $σ_{k}^{\frac{1}{k}}(λ(D_i (A\left(|Du|\right) D_j u)))\geq f(u)$ including the Laplacian, p-Laplacian, mean curvature, Hessian, k-mean curvature operators, and provide a necessary and sufficient condition on the global solvability, which can be regarded as generalized Keller-Osserman conditions.

math.DG↗
arXiv2022

Liouville property and existence of entire solutions of Hessian equations

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.

math.AP↗
arXiv2022

Asymptotic behavior of solutions to the Monge--Ampère equations with slow convergence rate at infinity

We consider the asymptotic behavior of solutions to the Monge--Ampère equations with slow convergence rate at infinity and fulfill previous results under faster convergence rate by Bao--Li--Zhang [Calc. Var PDE. 52(2015). pp. 39-63]. Different from known results, we obtain the limit of Hessian and/or gradient of solution at infinity relying on the convergence rate. The basic idea is to use a revised level set method, the spherical harmonic expansion and the iteration method.

math.AP↗
arXiv2022

Asymptotic expansion of 2-dimensional gradient graph with vanishing mean curvature at infinity

In this paper, we establish the asymptotic expansion at infinity of gradient graph in dimension 2 with vanishing mean curvature at infinity. This corresponds to our previous results in higher dimensions and generalizes the results for minimal gradient graph on exterior domain in dimension 2. Different from the strategies for higher dimensions, instead of the equivalence of Green's function on unbounded domains, we apply a version of iteration methods from Bao--Li--Zhang [Calc.Var PDE, 52(2015), pp. 39-63] that is refined by spherical harmonic expansions to provide a more explicit asymptotic behavior than known results.

math.AP↗
arXiv2021

Asymptotic expansion and optimal symmetry of minimal gradient graph equations in dimension 2

In this paper, we study asymptotic expansion at infinity and symmetry of zero mean curvature equations of gradient graph in dimension 2, which include the Monge--Ampère equation, inverse harmonic Hessian equation and the special Lagrangian equation. This refines the research of asymptotic behavior, gives the precise gap between exterior minimal gradient graph and the entire case, and extends the classification results of Monge--Ampère equations.

math.AP↗
arXiv2021

Pointwise A Priori Estimates for Solutions to Some p-Laplacian Equations

In this paper, we apply blow-up analysis and Liouville type theorems to study pointwise a priori estimates for some quasilinear equations with p-Laplace operator. We first obtain pointwise interior estimates for the gradient of p-harmonic function, i.e., the solution of $Δ_{p}u=0,\ x\inΩ$, which extends the well-established results of the interior estimates of the gradient of harmonic function. We then get singularity and decay estimates of the sign changing solution of Lane-Emden-Fowler type p-Laplace equation $-Δ_{p}u=|u|^{λ-1}u, \ x\inΩ$, which are then generalized for the equation with general right hand term $f(x,u)$, under some asymptotic conditions of $f$. Lastly, we get pointwise estimates for higher order derivatives of the solution of $-Δu=u^λ,x\inΩ$, the case of $p=2$ for p-Laplace equation.

math.AP↗
arXiv2021

Symmetry of solutions of minimal gradient graph equations on punctured space

In this paper, we study symmetry and existence of solutions of minimal gradient graph equations on punctured space $\mathbb R^n\setminus\{0\}$, which include the Monge-Ampère equation, inverse harmonic Hessian equation and the special Lagrangian equation. This extends the classification results of Monge-Ampère equations. Under some conditions, we also give the characterization of the solvability on exterior Dirichlet problem in terms of their asymptotic behaviors.

math.AP↗
arXiv2020

Asymptotic expansion at infinity of solutions of special Lagrangian equations

We obtain a quantitative high order expansion at infinity of solutions for a family of fully nonlinear elliptic equations on exterior domain, refine the study of the asymptotic behavior of the Monge-Ampère equation, the special Lagrangian equation and other elliptic equations, and give the precise gap between exterior maximal (or minimal) gradient graph and the entire case.

math.AP↗
arXiv2020

Asymptotic behavior at infinity of solutions of Lagrangian mean curvature equations

We studied the asymptotic behavior of solutions with quadratic growth condition of a class of Lagrangian mean curvature equations $F_τ(λ(D^2u))=f(x)$ in exterior domain, where $f$ satisfies a given asymptotic behavior at infinity. When f(x) is a constant near infinity, it is not necessary to demand the quadratic growth condition anymore. These results are a kind of exterior Liouville theorem, and can also be regarded as an extension of theorems of Pogorelov, Flanders and Yuan.

math.AP↗
arXiv2018

Existence of entire solutions of Monge-Ampère equations with prescribed asymptotic behaviors

We prove the existence of entire solutions of the Monge-Ampère equations with prescribed asymptotic behavior at infinity of the plane, which was left by Caffarelli-Li in 2003. The special difficulty of the problem in dimension two is due to the global logarithmic term in the asymptotic expansion of solutions at infinity. Furthermore, we give a PDE proof of the characterization of the space of solutions of the Monge-Ampère equation $\det \nabla^2 u=1$ with $k\ge 2$ singular points, which was established by Gálvez-Martínez-Mira in 2005. We also obtain the existence in higher dimensional cases with general right hand sides.

math.AP↗
arXiv2018

Liouville theorem and isolated singularity of fractional Laplacian system with critical exponents

This paper is devoted to the fractional Laplacian system with critical exponents. We use the method of moving sphere to derive a Liouville Theorem, and then prove the solutions in R^n\{0} are radially symmetric and monotonically decreasing radially. Together with blow up analysis and the Pohozaev integral, we get the upper and lower bound of the local solutions in B_1\{0}. Our results is an extension of the classical work by Caffarelli et al [6, 7], Chen et al[16]

math.AP↗
arXiv2017

Optimal boundary gradient estimates for Lamé systems with partially infinite coefficients

In this paper, we derive the pointwise upper bounds and lower bounds on the gradients of solutions to the Lamé systems with partially infinite coefficients as the surface of discontinuity of the coefficients of the system is located very close to the boundary. When the distance tends to zero, the optimal blow-up rates of the gradients are established for inclusions with arbitrary shapes and in all dimensions.

math.AP↗
arXiv2015

Global solutions and exterior Dirichlet problem for Monge-Ampere equation in $\mathbb R^2$

Monge-Ampère equation $\det(D^2u)=f$ in two dimensional spaces is different in nature from their counterparts in higher dimensional spaces. In this article we employ new ideas to establish two main results for the Monge-Ampère equation defined either globally in $\mathbb R^2$ or outside a convex set. First we prove the existence of a global solution that satisfies a prescribed asymptotic behavior at infinity, if $f$ is asymptotically close to a positive constant. Then we solve the exterior Dirichlet problem if data are given on the boundary of a convex set and at infinity.

math.AP↗