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On a regularity theorem for weak solutions to transmission problems with internal Lipschitz boundaries

Authors: L. Escauriaza, E. B. Fabes and G. Verchota
Journal: Proc. Amer. Math. Soc. 115 (1992), 1069-1076
MSC: Primary 35B65; Secondary 35J15
MathSciNet review: 1092919
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Abstract: We show that if $ u$ is a weak solution to $ \operatorname{div} (A\nabla u) = 0$ on an open set $ \Omega $ containing a Lipschitz domain $ D$, where $ A = kI{\chi _D} + I{\chi _{\Omega /D}}(k > 0,k \ne 1)$. Then, the nontangential maximal function of the gradient of $ u$ lies in $ {L^2}(\partial D)$.

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