Journal of Functional Analysis ( IF 1.7 ) Pub Date : 2021-04-23 , DOI: 10.1016/j.jfa.2021.109067 Derek W. Robinson
Let Ω be a domain in with boundary Γ, the Euclidean distance to the boundary and an elliptic operator with where are real, bounded, Lipschitz functions. We assume that as in the sense of asymptotic analysis where c is a strictly positive, bounded, Lipschitz function and . We also assume that there is an and a such that the weighted Hardy inequality is valid for all where . We then prove that the condition is sufficient for the essential self-adjointness of H on with the supremum over r of all possible in the Hardy inequality. This result extends all known results for domains with smooth boundaries and also gives information on self-adjointness for a large family of domains with rough, e.g. fractal, boundaries.
中文翻译:
对称扩散算子的加权Hardy不等式和自伴性
设Ω为 边界为Γ 到边界的欧几里得距离和 一个椭圆运算符 在哪里 是真实的,有界的Lipschitz函数。我们假设 作为 在渐近分析的意义上,其中c是严格的正有界Lipschitz函数,。我们还假设存在一个 和一个 这样加权的Hardy不等式 对所有人都有效 在哪里 。然后我们证明条件足以满足H on的基本自伴性 和 r在所有可能之上的最高在哈代不平等中。该结果扩展了具有平滑边界的域的所有已知结果,并且还提供了具有粗糙(例如分形)边界的大量域的自伴随性信息。