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Oct 5, 2021 at 3:03 comment added user378654 You are correct and I misread part of the question. However, I believe essentially the same argument still applies (solve for the harmonic function first, then extend), and I have modified the answer.
Oct 5, 2021 at 3:01 history edited user378654 CC BY-SA 4.0
Correction for Michael Renardy's comment below.
Oct 5, 2021 at 2:02 comment added Michael Renardy But why is ${\rm supp}(f)\subset D_2\backslash\overline{D_1}$?
Oct 4, 2021 at 18:31 vote accept Ali
Oct 4, 2021 at 18:25 history answered user378654 CC BY-SA 4.0