Monge-Ampère方程边界爆破解的最优估计和不存在性

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On the Optimal Global Estimates of Boundary Blow-up Solutions to the Monge-Ampère Equation

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Abstract:

This paper is dedicated to studying the optimal global estimates and nonexistence of strictly convex solutions to the boundary blow-up Monge-Ampère problem $M[u](x)=K(x)f(u) \mbox{ for } x \in \Omega,\; u(x)\rightarrow +\infty \mbox{ as } {\rm dist}(x,\partial \Omega)\rightarrow 0.$ Here $M[u]=\det\, (u_{x_{i}x_{j}})$ is the Monge-Ampère operator, and $\Omega$ denotes a smooth, bounded, strictly convex domain in $\Bbb R^N (N\geq 2)$ . The interesting features in our proof are that we not only obtain the relations among various conditions imposed on $K(x)$ and $f(u)$ , but make comparison of some results of global estimates in previous literatures and make clear what conditions lead to what estimations. Moreover, when $\Omega$ is a general region, we give some nonexistence results which is rarely discussed in previous literatures.

Key words: Monge-Ampère equation, Boundary blow-up, Global estimates, Strictly convex solution, Nonexistence

CLC Number:

O177.91

Feng Meiqiang, Zhang Xuemei. On the Optimal Global Estimates of Boundary Blow-up Solutions to the Monge-Ampère Equation[J].Acta mathematica scientia,Series A, 2023, 43(1): 181-202.

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