Assume that S2 is a bounded open set in R2 and that g: 012 → R continuous, with the following ball boundary condition: f

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Assume That S2 Is A Bounded Open Set In R2 And That G 012 R Continuous With The Following Ball Boundary Condition F 1 (184.71 KiB) Viewed 33 times

Assume that S2 is a bounded open set in R2 and that g: 012 → R continuous, with the following ball boundary condition: for any x € 212, there is an open ball B with xe aB and BC 12. (1) When c € 12 and R > 0 are the center and the radius of the latter ball, we then denote for any u € C'() ди u(x) – u(x – €(x – c))) (x) = lim ay 8|x - cl We consider the boundary value ди, Au = 0, Vx € 12 and 4(x) = g(x), Vx E 29. (*) ay (a) Give an example of a domain 12 satisfying the condition (1) for all x € 222 but that is not C!. (b) If u, v E C?(9) nC'n) are two solutions of (*), show that u = v + C for some constant C. Hint: You may find useful to use Hopf Lemma.