AI 中文总结
本文针对一致椭圆方程的三个经典问题，提出尖锐梯度界并改进特征值渐近与Landis猜想，统一方法结合Harnack不等式、对偶与重标度，以反例证明最优性。
AI 中文摘要
我们研究线性二阶一致椭圆方程理论中的三个经典问题：(i) Dirichlet问题的梯度估计，(ii) 椭圆算子第一特征值的估计与渐近行为，以及(iii) 关于指数衰减的Landis猜想。我们的主要结果给出了(i)的若干版本，其中常数以算子系数的范数和区域的大小被精确指定；并利用这些结果，通过允许更一般的算子和对系数更弱的正则性假设，以及给出定量估计，大幅改进了关于(ii)和(iii)的已知结果。证明采用统一方法，依赖于三个要素：具有尖锐常数的内部与边界Harnack不等式、对偶论证，以及基于重标度过程的$C^1$估计。梯度估计和谱估计的尖锐性通过多种反例得到验证。
英文摘要
We study three classical problems in the theory of linear second-order uniformly elliptic equations: (i) gradient estimates for the Dirichlet problem, (ii) estimates and asymptotics for the first eigenvalue of an elliptic operator, and (iii) the Landis conjecture on exponential decay. Our main results give versions of (i) in which the constant is sharply specified in terms of the norms of the coefficients of the operator and the size of the domain; and use these to strongly improve on known results for (ii) and (iii) by allowing both more general operators and weaker regularity assumptions on the coefficients, and by giving quantitative estimates. The proofs use a unified approach, relying on three ingredients: interior and boundary Harnack inequalities with sharp constants, duality arguments, and a $C^1$ estimate based on a rescaling procedure. The sharpness of the gradient and spectral estimates is demonstrated through various counterexamples.
CommentsThis preprint stems from preprint ArXiV 2411.19367, which was split in two parts, and whose results have been substantially improved and extended
