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Assume that is a bounded open set in R2 and that g: dR continuous, with the following ball boundary condition: for any xan, there is an open ball B with x € 9B and B C Ω. (1) When cand R> 0 are the center and the radius of the latter ball, we then denote for any и є C'(§) ди -(x) = lim ἂν 8-0 We consider the boundary value u(x) - u(x − x(x - c))) Elx - cl Δu = 0, Vx € Ω and ди -(x) = g(x), Vx € an. ὃν (*) (a) Give an example of a domain satisfying the condition (1) for all x € N but that is not C¹. (b) If u, v € C²(2) C'() are two solutions of (*), show that u = v + C for some constant C. Hint: You may find useful to use Hopf Lemma.

Assume that is a bounded open set in R2 and that g: dR continuous, with the following ball boundary condition: for any xan, there is an open ball B with x € 9B and B C Ω. (1) When cand R> 0 are the center and the radius of the latter ball, we then denote for any и є C'(§) ди -(x) = lim ἂν 8-0 We consider the boundary value u(x) - u(x − x(x - c))) Elx - cl Δu = 0, Vx € Ω and ди -(x) = g(x), Vx € an. ὃν (*) (a) Give an example of a domain satisfying the condition (1) for all x € N but that is not C¹. (b) If u, v € C²(2) C'() are two solutions of (*), show that u = v + C for some constant C. Hint: You may find useful to use Hopf Lemma.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Assume that Q is a bounded open set in R² and that g : aN → R continuous, with the following ball boundary condition: for any x e aN, there is an open ball B with хе дB аnd B сQ. (1) When c e N and R > 0 are the center and the radius of the latter ball, we then denote for any u e C'(N) ди (x) = lim dv и(х) — и(х — 8(х — с))) ɛ|x – c| We consider the boundary value ди Δυ-0, YXΕΩ md (1) -g), Vx ε 00. dv (*) (a) Give an example of a domain Q satisfying the condition (1) for all x e aN but that is not C'. (b) If u, v e C²N) n C'(N) are two solutions of (*), show that u = v + C for some constant C. Hint: You may find useful to use Hopf Lemma.
Transcribed Image Text:Assume that Q is a bounded open set in R² and that g : aN → R continuous, with the following ball boundary condition: for any x e aN, there is an open ball B with хе дB аnd B сQ. (1) When c e N and R > 0 are the center and the radius of the latter ball, we then denote for any u e C'(N) ди (x) = lim dv и(х) — и(х — 8(х — с))) ɛ|x – c| We consider the boundary value ди Δυ-0, YXΕΩ md (1) -g), Vx ε 00. dv (*) (a) Give an example of a domain Q satisfying the condition (1) for all x e aN but that is not C'. (b) If u, v e C²N) n C'(N) are two solutions of (*), show that u = v + C for some constant C. Hint: You may find useful to use Hopf Lemma.
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