Question

QUESTION 3 Show that the Neumann problem = 9 on a. vu = 0 in 2, ди an has a unique solution only up to an additive constant, i.e. if u and un are two solutions, then U1 - U2 =C. Hint: Use the identity div (uVu) = Vul? + uvºu, 2 ди where Vul ar vector field u Vu. And then apply the Gauss' Divergence Theorem to the

Answer 1

To show a The Meumann problem x?u= in a du onda on ² g has unique solution only upto an additive Constant of m & ų are two solurias, solutions, then -ų=c Proof? Suppose that ų and ų Solve the Heremann Problem for harmonic function, it Auzo in on -g an ar put W z uy-ų then Solves the homogeneous Herremann problem AW z a ih r ow 2 o in or an An application of Green] Green} first identify makes it plains that SS ow ds a on to ISO X w. awar SSS wande 2

Since, aw и от гя. В Дъно мя we see that О. JJ I w. Tanda - || | ~~ча. Since the integrando Hwly It follows that buy the vanishing theorem that у го Я е. TW=0 п, and we dedree that in constant in s say w=c Then, By our Construction, me se mat и – «, с 4 5 а ) и, - 41с & we «Алма или of the Лелл Problem нь constant required. 24 ищу thro sono 4-та Elence, proved)