Assume that 2 is a bounded open set in R2 and that g: 092 → R continuous, with the following ball boundary condition: fo

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Assume That 2 Is A Bounded Open Set In R2 And That G 092 R Continuous With The Following Ball Boundary Condition Fo 1 (224.45 KiB) Viewed 33 times

Assume that 2 is a bounded open set in R2 and that g: 092 → R continuous, with the following ball boundary condition: for any x € 212, there is an open ball B with xe aB and B C 12. (1) When c E N and R > 0 are the center and the radius of the latter ball, we then denote for any u € C'@ au u(x) – u(x – 8(x – c))) -(x) = lim ay 60 Elx - cl We consider the boundary value ди Au = 0, Vx € S and (x) = g(x), Vx € DS2. ) = (). (*) (a) Give an example of a domain 2 satisfying the condition (1) for all x € 292 but that is not C! (b) If u,v e C?(12) n c'@) are two solutions of (*), show that u = v + C for some constant C. Hint: You may find useful to use Hopf Lemma.