超定自由边界问题的联合非对称扰动法
The simultaneous asymmetric perturbation method for overdetermined free boundary problems
摘要：
本文介绍了一种应用隐函数定理寻找具有固定边界（给定）和自由边界（待定）的超定问题的非平凡解的新方法。这种方法的新颖之处在于所考虑的扰动类型。事实上，我们处理的扰动在每个边界上表现出不同的正则性水平。这使我们能够构建否则无法实现的解决方案。该方法的另一个优点在于改善了自由边界与给定边界之间的正则性差距。最后讨论了解的一些几何性质，如对称性和凸性。
英文摘要：
In this paper, we introduce a new method for applying the implicit function theorem to find nontrivial solutions to overdetermined problems with a fixed boundary (given) and a free boundary (to be determined). The novelty of this method lies in the kind of perturbations considered. Indeed, we work with perturbations that exhibit different levels of regularity on each boundary. This allows us to construct solutions that would have been out of reach otherwise. Another benefit of this method lies in the improvement of the regularity gap between the free boundary and the given one. Finally, some geometric properties of the solutions, such as symmetry and convexity, are also discussed.
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