Generalized versus classical normal derivative
DOI:
https://doi.org/10.24193/subbmath.2023.1.02Keywords:
Lipschitz domain, normal derivative, Green formula, generalized normal derivative, Neumann problem, strong maximum principle.Abstract
Given a bounded domain with Lipschitz boundary, the general Green formula permits to justify that the weak solutions of a Neumann elliptic problem satisfy the Neumann boundary condition in a weak sense. The formula involves a generalized normal derivative. We prove a general result which establishes that the generalized normal derivative of an operator coincides with the classical one, provided that the operator is continuous. This result allows to deduce that, under usual regularity assumptions, the weak solutions of a Neumann problem satisfy the Neumann boundary condition in the classical sense. This information is necessary in particular for applying the strong maximum principle.
Mathematics Subject Classification (2010): 26B20, 46E35, 46T20, 35J25, 35B50.
Received 01 October 2022; Revised 01 February 2023. Published Online: 2023-03-20. Published Print: 2023-04-30
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