Mirror symmetry for a Hessian over-determined problem and its generalization
Bo Wang Jiguang Bao
Communications on Pure & Applied Analysis 2014, 13(6): 2305-2316 doi: 10.3934/cpaa.2014.13.2305
In the paper, we apply the moving plane method to prove that if the right hand sides of equation and Neumann boundary condition are both independent of one variable, the domain and the solution to the Hessian over-determined problem are mirror symmetric. Our result generalizes the previous results on radial symmetry. In the end, we get the mirror symmetry of over-determined problems for more general equations, which include Weingarten curvature equation.
keywords: $k-$Hessian equation moving plane method Mirror symmetry corner lemma. over-determined problem Weingarten curvature equation
Euler-Poisson equations related to general compressible rotating fluids
Haigang Li Jiguang Bao
Discrete & Continuous Dynamical Systems - A 2011, 29(3): 1085-1096 doi: 10.3934/dcds.2011.29.1085
This paper is mainly concerned with Euler-Poisson equations modeling Newtonian stars. We establish the existence of rotating star solutions for general compressible fluids with prescribed angular velocity law, which is the main point distinguished with the case with prescribed angular momentum per unit mass. The compactness of any minimizing sequence is established, which is important from the stability point of view.
keywords: Euler-Poisson equations White dwarfs. Given angular velocity Rotating star
The obstacle problem for Monge-Ampère type equations in non-convex domains
Jingang Xiong Jiguang Bao
Communications on Pure & Applied Analysis 2011, 10(1): 59-68 doi: 10.3934/cpaa.2011.10.59
In this paper, we consider the obstacle problem for Monge-Ampère type equations which include prescribed Gauss curvature equation as a special case. We establish $C^{1,1}$ regularity of the greatest viscosity solution in non-convex domains.
keywords: non-convex domain. Monge-Ampère equation Obstacle problem
New maximum principles for fully nonlinear ODEs of second order
Wenmin Sun Jiguang Bao
Discrete & Continuous Dynamical Systems - A 2007, 19(4): 813-823 doi: 10.3934/dcds.2007.19.813
In the paper, we give the positive answer of an open problem of Li-Nirenberg under the weaker conditions, and we prove a new variation of the boundary point lemma for second order fully nonlinear ODEs by a new method. A simpler proof of Li-Nirenberg Theorem is also presented.
keywords: boundary point lemma. fully nonlinear Li-Nirenberg open problem
Polyharmonic equations with critical exponential growth in the whole space $ \mathbb{R}^{n}$
Jiguang Bao Nguyen Lam Guozhen Lu
Discrete & Continuous Dynamical Systems - A 2016, 36(2): 577-600 doi: 10.3934/dcds.2016.36.577
In this paper, we apply the sharp Adams-type inequalities for the Sobolev space $W^{m,\frac{n}{m}}\left( \mathbb{R} ^{n}\right) $ for any positive real number $m$ less than $n$, established by Ruf and Sani [46] and Lam and Lu [30,31], to study polyharmonic equations in $\mathbb{R}^{2m}$. We will consider the polyharmonic equations in $\mathbb{R}^{2m}$ of the form \[ \left( I-\Delta\right) ^{m}u=f(x,u)\text{ in }% \mathbb{R} ^{2m}. \] We study the existence of the nontrivial solutions when the nonlinear terms have the critical exponential growth in the sense of Adams' inequalities on the entire Euclidean space. Our approach is variational methods such as the Mountain Pass Theorem ([5]) without Palais-Smale condition combining with a version of a result due to Lions ([39,40]) for the critical growth case. Moreover, using the regularity lifting by contracting operators and regularity lifting by combinations of contracting and shrinking operators developed in [14] and [11], we will prove that our solutions are uniformly bounded and Lipschitz continuous. Finally, using the moving plane method of Gidas, Ni and Nirenberg [22,23] in integral form developed by Chen, Li and Ou [12] together with the Hardy-Littlewood-Sobolev type inequality instead of the maximum principle, we prove our positive solutions are radially symmetric and monotone decreasing about some point. This appears to be the first work concerning existence of nontrivial nonnegative solutions of the Bessel type polyharmonic equation with exponential growth of the nonlinearity in the whole Euclidean space.
keywords: symmetry and regularity of solutions existence Bessel potential Polyharmonic equations Moser-Trudinger and Adams inequalities. exponential growth

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