星形域共形参数化与半线性椭圆解的持久性
Conformal Parametrisation of Star-Shaped Domains and Persistence of Semilinear Elliptic Solutions
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中文总结 AI 辅助
研究星形域共形参数化，通过Wiener-Hopf分解和计算机辅助估计验证无限维星形域族，并证明半线性椭圆解的持久性。
中文摘要 AI 辅助
我们研究了光滑星形平面域的共形参数化及其对边界扰动的依赖性。归一化黎曼映射的边界对应关系由Theodorsen方程描述，我们在加权Wiener代数中表述该方程。通过典型的Wiener-Hopf分解，我们推导了线性化Theodorsen算子逆的显式表示，并获得了在边界函数扰动下稳定的定量后验估计。对于一组非微扰参考域，严格的计算机辅助估计在加权Wiener代数中给出了显式邻域，使得这些邻域中的每个边界函数在验证球内都具有唯一确定的共形参数化，并对相应的黎曼映射进行定量控制。因此，该验证适用于开集、无限维的星形域族。作为应用，我们考虑半线性Dirichlet问题 \\[-\Delta v=v^3\quad\text{in }\Omega, \qquad v=0\quad\text{on }\partial\Omega.\\] 通过验证的共形映射将方程拉回到单位圆盘上，得到一个单位圆盘上的问题。一致的后验估计随后暗示椭圆解在参考域的显式邻域内具有持久性。
英文摘要
We study the conformal parametrisation of smooth star-shaped planar domains and its dependence on perturbations of the boundary. The boundary correspondence of the normalized Riemann map is described by the Theodorsen equation, which we formulate in weighted Wiener algebras. We derive an explicit representation of the inverse of the linearized Theodorsen operator through a canonical Wiener--Hopf factorisation and obtain quantitative a posteriori estimates that are stable under perturbations of the boundary function. For a collection of nonperturbative reference domains, rigorous computer-assisted estimates yield explicit neighborhoods in a weighted Wiener algebra such that every boundary function in these neighborhoods admits a uniquely determined conformal parametrisation in the validated ball, with quantitative control of the corresponding Riemann map. Thus the validation applies to open, infinite-dimensional families of star-shaped domains. As an application, we consider the semilinear Dirichlet problem \[ -Δv=v^3\quad\text{in }Ω, \qquad v=0\quad\text{on }\partialΩ. \] Pulling the equation back by the validated conformal maps yields a problem on the unit disk. Uniform a posteriori estimates then imply persistence of the elliptic solutions throughout explicit neighborhoods of the reference domains.
发表机构
- Politecnico di Milano（米兰理工大学）
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