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192 results on '"Zhang, Weijun"'

1. Symmetry of Convex Solutions to Fully Nonlinear Elliptic Systems: Bounded Domains

Author

Zhang, Weijun and Zhang, Zhitao

Subjects
Mathematics - Analysis of PDEs, Mathematics - Functional Analysis
Abstract

In this paper, we are concerned with the monotonic and symmetric properties of convex solutions to fully nonlinear elliptic systems. We mainly discuss Monge-Amp\`ere type systems for instance, considering \begin{equation*} \det(D^2u^i)=f^i(x,{\bf u},\nabla u^i), \ 1\leq i\leq m, \end{equation*} over bounded domains of various cases, including the bounded smooth simply connected domains and bounded tube shape domains in $\mathbb{R}^n$. We obtain monotonic and symmetric properties of the solutions to the problem with respect to the geometry of domains and the monotonic and symmetric properties of right-hand side terms. The proof is based on carefully using the moving plane method together with various maximum principles and Hopf's lemmas. The existence and uniqueness to an interesting example of such system is also discussed as an application of our results., Comment: comments are welcome! arXiv admin note: substantial text overlap with arXiv:2404.01185

Published
2024

2. Bifurcation on Fully Nonlinear Elliptic Equations and Systems

Author

Gao, Jing, Zhang, Weijun, and Zhang, Zhitao

Subjects
Mathematics - Analysis of PDEs, Mathematics - Functional Analysis
Abstract

In this paper, we study the following fully nonlinear elliptic equations \begin{equation*} \left\{\begin{array}{rl} \left(S_{k}(D^{2}u)\right)^{\frac1k}=\lambda f(-u) & in\quad\Omega \\ u=0 & on\quad \partial\Omega\\ \end{array} \right. \end{equation*} and coupled systems \begin{equation*} \left\{\begin{array}{rl} (S_{k}(D^{2}u))^\frac1k=\lambda g(-u,-v) & in\quad\Omega \\ (S_{k}(D^{2}v))^\frac1k=\lambda h(-u,-v) & in\quad\Omega \\ u=v=0 & on\quad \partial\Omega\\ \end{array} \right. \end{equation*} dominated by $k$-Hessian operators, where $\Omega$ is a $(k$-$1)$-convex bounded domain in $\mathbb{R}^{N}$, $\lambda$ is a non-negative parameter, $f:\left[0,+\infty\right)\rightarrow\left[0,+\infty\right)$ is a continuous function with zeros only at $0$ and $g,h:\left[0,+\infty\right)\times \left[0,+\infty\right)\rightarrow \left[0,+\infty\right)$ are continuous functions with zeros only at $(\cdot,0)$ and $(0,\cdot)$. We determine the interval of $\lambda$ about the existence, non-existence, uniqueness and multiplicity of $k$-convex solutions to the above problems according to various cases of $f,g,h$, which is a complete supplement to the known results in previous literature. In particular, the above results are also new for Laplacian and Monge-Amp\`ere operators. We mainly use bifurcation theory, a-priori estimates, various maximum principles and technical strategies in the proof., Comment: comments are welcome!

Published
2024

3. Symmetry of Convex Solutions to Fully Nonlinear Elliptic Systems: Unbounded Domains

Author

Zhang, Weijun and Zhang, Zhitao

Subjects
Mathematics - Analysis of PDEs, Mathematics - Functional Analysis
Abstract

In this paper, we are concerned with the monotonic and symmetric properties of convex solutions Monge-Amp\`ere systems for instance, considering \begin{equation*} \det(D^2u^i)=f^i(x,{\bf u},\nabla u^i), \ 1\leq i\leq m, \end{equation*} over unbounded domains of various cases, including the whole spaces $\mathbb{R}^n$, the half spaces $\mathbb{R}^n_+$ and the unbounded tube shape domains in $\mathbb{R}^n$. We obtain monotonic and symmetric properties of the solutions to the problem with respect to the geometry of domains and the monotonic and symmetric properties of right-hand side terms. The proof is based on carefully using the moving plane method together with various maximum principles and Hopf's lemmas., Comment: comments are welcome!

Published
2024

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