(a) The function u satisfies V2u = 0 in the volume V and u = 0...

 

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(a) The function u satisfies V2u = 0 in the volume V and u = 0 on S, the surface bounding V. Show that u = 0 everywhere in V. The function u satisfies V² = 0 in V and v is specified on S. Show that for all functions w such that w = v on S √ Do Vv. VwdV = ₁ v² dv. Hence show that √₁₂ w²dV = ₁ {\v² + 1 (w_v) ³}dV > ✓ \\v² dv. (b) The function satisfies V2 = p(x) in the spherical region |x| <a, with = 0 on xa. The function p(x) is spherically symmetric, i.e. p(x) = p(x) = p(r). Suppose that the equation and boundary conditions are satisfied by a spherically symmetric function (r). Show that Anr²4 (r) = 4ns²p(s) ds. Hence find the function Þ(r) when p(r) is given by p(r) with po constant. Explain how the results obtained in part (a) of the question imply that (r) is the only solution of V² = p(r) which satisfies the specified boundary condition on |x| = <= a. Ju(a)! where U(b, a) is the region b<r <a. = Use your solution and the results obtained in part (a) of the question to show that, for any function w such that w=1 on r = b and w= 0 on r = a, |vw|²dV > [ po if 0 <r<b 10_ifb<r <a' Arab a-b' (a) The function u satisfies V2u = 0 in the volume V and u = 0 on S, the surface bounding V. Show that u = 0 everywhere in V. The function u satisfies V² = 0 in V and v is specified on S. Show that for all functions w such that w = v on S √ Do Vv. VwdV = ₁ v² dv. Hence show that √₁₂ w²dV = ₁ {\v² + 1 (w_v) ³}dV > ✓ \\v² dv. (b) The function satisfies V2 = p(x) in the spherical region |x| <a, with = 0 on xa. The function p(x) is spherically symmetric, i.e. p(x) = p(x) = p(r). Suppose that the equation and boundary conditions are satisfied by a spherically symmetric function (r). Show that Anr²4 (r) = 4ns²p(s) ds. Hence find the function Þ(r) when p(r) is given by p(r) with po constant. Explain how the results obtained in part (a) of the question imply that (r) is the only solution of V² = p(r) which satisfies the specified boundary condition on |x| = <= a. Ju(a)! where U(b, a) is the region b<r <a. = Use your solution and the results obtained in part (a) of the question to show that, for any function w such that w=1 on r = b and w= 0 on r = a, |vw|²dV > [ po if 0 <r<b 10_ifb<r <a' Arab a-b'