与$p$-Laplacian相关的正则化两相问题的分岔现象
The Bifurcation Phenomenon for the Regularized Two-Phase Problem Associated with the $p$-Laplacian
- University of Michigan（密歇根大学）
- Wayne State University（韦恩州立大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文证明了与$p$-Laplacian相关的正则化两相自由边界问题在边界数据较小时存在山隘点解，从而产生三个弱解，并通过径向对称情形下的常微分方程揭示了自由边界Bernoulli条件导致的分岔现象。
AI中文摘要：
本文研究了在Dirichlet边界条件下，与$p$-Laplacian相关的正则化两相自由边界问题弱解多重性的分岔现象。事实上，我们证明了当边界数据较小时，存在一个山隘点解。在这种情况下，正则化问题存在三个弱解：$p$-调和函数、相应泛函的极小元以及山隘点解。最后，我们考虑了径向对称解的特殊情形。通过在非正则化情形下求解相关的非线性常微分方程，我们明确展示了自由边界处的Bernoulli条件如何导致解的分岔。
英文摘要：
In this paper, we verify a bifurcation phenomenon regarding the multiplicity of weak solutions, subject to the Dirichlet boundary condition, of a regularized two-phase free boundary problem associated with the $p$-Laplacian. In fact, we prove the existence of a mountain pass solution when the boundary data is small. In this case, three weak solutions of the regularized problem exist: the $p$-harmonic function, a minimizer of the corresponding functional, and the mountain pass solution. Finally, we consider the special case of radially symmetric solutions. By solving the associated nonlinear ordinary differential equation in the unregularized case, we show explicitly how the Bernoulli condition at the free boundary gives rise to the bifurcation of solutions.
