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The Vázquez maximum principle and the Landis conjecture for elliptic PDE with unbounded coefficients
Advances in Mathematics ( IF 1.5 ) Pub Date : 2021-06-18 , DOI: 10.1016/j.aim.2021.107838
Boyan Sirakov , Philippe Souplet

We develop a new, unified approach to the following two classical questions on elliptic PDE:

the strong maximum principle for equations with non-Lipschitz nonlinearities,

the at most exponential decay of solutions in the whole space or exterior domains.

Our results apply to divergence and non-divergence operators with locally unbounded lower-order coefficients, in a number of situations where all previous results required bounded ingredients. Our approach, which allows for relatively simple and short proofs, is based on a (weak) Harnack inequality with optimal dependence of the constants in the lower-order terms of the equation and the size of the domain, which we establish.



中文翻译:

具有无界系数的椭圆偏微分方程的 Vázquez 最大值原理和兰迪斯猜想

我们针对椭圆偏微分方程的以下两个经典问题开发了一种新的、统一的方法:

非Lipschitz非线性方程的强最大值原理,

整个空间或外部域中解的至多指数衰减。

我们的结果适用于具有局部无界低阶系数的发散和非发散算子,在许多情况下,所有先前的结果都需要有界成分。我们的方法允许相对简单和简短的证明,它基于(弱)Harnack 不等式,具有方程低阶项中的常数和我们建立的域大小的最佳依赖关系。

更新日期:2021-06-18
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