论文标题
elouville定理和椭圆方程中的最佳规律性
Liouville theorems and optimal regularity in elliptic equations
论文作者
论文摘要
本文的目的是建立与可测量系数的椭圆形PDE的最佳规律性问题与无穷大的liouville属性之间的联系。最初，我们通过证明Alt-Caffarelli-Friedman型单调性公式来解决二维案例，从而实现了最佳规律性证明和用于多相问题的Liouville属性。在较高的维度中，我们深入研究单调公式在表征最佳规律性方面的作用。通过采用填充技术，我们提出了一种独特的“几乎单调性”公式，暗示了h $Ö$ $ lder的解决方案。最后，我们通过结合爆炸和$ g $ convergence的论点来探讨无穷大增长与规律性指数之间的相互作用。
The objective of this paper is to establish a connection between the problem of optimal regularity among solutions to elliptic PDEs with measurable coefficients and the Liouville property at infinity. Initially, we address the two-dimensional case by proving an Alt-Caffarelli-Friedman type monotonicity formula, enabling the proof of optimal regularity and the Liouville property for multiphase problems. In higher dimensions, we delve into the role of monotonicity formulas in characterizing optimal regularity. By employing a hole-filling technique, we present a distinct "almost-monotonicity" formula that implies H$ö$lder regularity of solutions. Finally, we explore the interplay between the least growth at infinity and the exponent of regularity by combining blow-up and $G$-convergence arguments.
