椭圆型PDE节点解集的广义光滑性与Chojasiewicz不等式
Łojasiewicz Inequalities and Generic Smoothness of Nodal Sets of Solutions to Elliptic PDE
摘要：
在这篇文章中，我们证明了一个广泛的二阶椭圆偏微分方程，包括拉普拉斯算子，零集的狄利克雷问题的解决方案是光滑的“通用”$L^2$数据。当解的零点集（例如调和函数）包含奇点时，这意味着我们可以找到边界数据的任意小扰动，使得扰动解的零点集在前一个奇点的指定邻域内是光滑的。此外，我们可以把微扰取为“平均零”，这就有了额外的技术困难，以确保我们在消除原始奇点的过程中不会引入新的奇点。独立的兴趣，为了证明的主要定理，我们建立了一个有效的版本的梯度不等式的一致常数在类的解有界频率。
英文摘要：
In this article, we prove that for a broad class of second order elliptic PDEs, including the Laplacian, the zero sets of solutions to the Dirichlet problem are smooth for "generic" $L^2$ data. When the zero set of a solution (e.g. a harmonic function) contains a singularity, this means that we can find an arbitrarily small perturbation of the boundary data so that the zero set of the perturbed solution is smooth throughout a prescribed neighborhood of the former singularity. Furthermore, we can take the perturbation to be "mean zero" for which there are additional technical difficulties to ensure that we do not introduce new singularities in the process of eliminating the original ones. Of independent interest, in order to prove the main theorem, we establish an effective version of the Łojasiewicz gradient inequality with uniform constants in the class of solutions with bounded frequency.
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