Translated Abstract
This thesis discussed the global asymptotic behaviour of one phase free boundary problem in some specific exterior domain. We assumed the boundary of the exterior domain is compactly supported in positive phase, and that the free boundary is a smooth graph. We adapted some classical tools for free boundary problems with some possible changes, including blow-down technique and Weiss＇s monotonicity formula, to this situation, and obtained some global asymptotic behaviour results.
In the second chapter, we introduced one phase free boundary problem, including its derivation, important related facts and classical tools. The study on the local regularity and global asymptotic behaviour has been emphasized.
In the third chapter, we dealt with the exterior problem for one phase free boundary problem in general R^n(n＞= 2). Specifically, we considered the case when the boundary of exterior domain is compactly supported in the positive phase, and for convenience we assume the free boundary is a smooth graph. For the sake of calculation, in the later part of this thesis we mainly consider one of its equivalent form. We proved the linear growth of the solution, i.e., it＇s Lipschitz continuous and non-degenerate. We added some extra term to the Weiss＇s monotonicity formula, to ensure that monotonicity still hold in large. Then, we characterize the blow-down limit of u, and proved it to be a one-degree homogeneous global variational solution.
In the fourth chapter, we explored the planar case in detail. We proved that the blow-down limit in this case is indeed linear. Thus it＇s evident that $u$ is asymptotically flat in the subsequence sense.
In the fifth chapter, we summarized the content of this article and mentioned some corresponding problems one could consider next.
Translated Keyword
[Asymptotic behaviour, Exterior domain problem, Free boundary problem, Partial differential equations]
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