发表机构
Sapienza Università di Roma; Tsuda University（罗马大学; 津田塾大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对$p>1$的 fully nonlinear degenerate elliptic equations边值问题，结合先验Lipschitz估计，分析解的存在性、渐近行为及退化算子与哈密顿量的相互作用，揭示不同$i$值下的解的现象。
AI 中文摘要
对于$p>1$，我们考虑有界域内 fully nonlinear degenerate elliptic equations $-\lambda_i(D^2u)+|Du|^p+\gamma u=f(x)$的边值问题，其边界条件为Dirichlet条件或边界爆破条件；此处$\lambda_i(D^2u)$表示Hessian的第$i$个特征值。我们研究解的存在性与不存在性，以及当$\gamma$趋于0时解的渐近行为。先验Lipschitz估计发挥重要作用。算子的退化性与哈密顿量的超线性增长之间的相互作用会产生截然不同的现象，具体取决于哪一项占主导地位，例如遍历二分仅在$i=N$时发生，而对于$i<N$则会出现新现象：在温和条件下，仅在单点爆破的解不存在，且齐次边界条件的Dirichlet问题的解的存在性需对$f$的大小施加条件。
英文摘要
For $p>1$, we consider the boundary value problem for fully nonlinear degenerate elliptic equations $-λ_i(D^2u)+|Du|^p+γu=f(x)$ in bounded domains with Dirichlet or boundary blow-up conditions; here $λ_i(D^2u)$ denotes the $i$-th eigenvalue of the Hessian. We study existence and nonexistence of solutions together with the asymptotic behaviour of the solutions when $γ$ goes to zero. A priori Lipschitz estimates play an important role. The interplay between the operator's degeneracy and the superlinear growth of the Hamiltonian gives rise to phenomena that are very different depending on which of the two terms dominates, e.g. the ergodic dichotomy takes place only when $i=N$, while new phenomena arise for $i<N$ in which case, under mild conditions, solutions that blow up even in just one point do not exist, and conditions on the size of $f$ must be imposed for the existence of solutions to the Dirichlet problem with homogeneous boundary condition.
Comments48 pages
