Mathematical Methods for Physics and Engineering - Matematica.NET

PDES: GENERAL AND PARTICULAR SOLUTIONSAs an important example let us consider Poisson’s equation in three dimensions,∇ 2 u(r) =ρ(r), (20.46)with either Dirichlet or Neumann conditions on a closed boundary appropriateto such an elliptic equation; for brevity, in (20.46), we have absorbed any physicalconstants into ρ. We aim to show that, to within an unimportant constant, thesolution of (20.46) is unique if either the potential u or its normal derivative∂u/∂n is specified on all surfaces bounding a given region of space (including, ifnecessary, a hypothetical spherical surface of indefinitely large radius on which uor ∂u/∂n is prescribed to have an arbitrarily small value). Stated more formallythis is as follows.Uniqueness theorem. If u is real and its first and second partial derivatives arecontinuous in a region V and on its boundary S, and ∇ 2 u = ρ in V and eitheru = f or ∂u/∂n = g on S, whereρ, f and g are prescribed functions, then u isunique (at least to within an additive constant).◮Prove the uniqueness theorem for Poisson’s equation.Let us suppose on the contrary that two solutions u 1 (r)andu 2 (r) both satisfy the conditionsgiven above, and denote their difference by the function w = u 1 − u 2 . We then have∇ 2 w = ∇ 2 u 1 −∇ 2 u 2 = ρ − ρ =0,so that w satisfies Laplace’s equation in V . Furthermore, since either u 1 = f = u 2 or∂u 1 /∂n = g = ∂u 2 /∂n on S, we must have either w =0or∂w/∂n =0onS.If we now use Green’s first theorem, (11.19), for the case where both scalar functionsare taken as w we have ∫[w∇ 2 w +(∇w) · (∇w) ] ∫dV = w ∂wVS ∂n dS.However, either condition, w =0or∂w/∂n = 0, makes the RHS vanish whilst the firstterm on the LHS vanishes since ∇ 2 w =0inV . Thus we are left with∫|∇w| 2 dV =0.VSince |∇w| 2 can never be negative, this can only be satisfied if∇w = 0,i.e. if w, and hence u 1 − u 2 , is a constant in V .If Dirichlet conditions are given then u 1 ≡ u 2 on (some part of) S and hence u 1 = u 2everywhere in V . For Neumann conditions, however, u 1 and u 2 can differ throughout Vby an arbitrary (but unimportant) constant. ◭The importance of this uniqueness theorem lies in the fact that if a solution toPoisson’s (or Laplace’s) equation that fits the given set of Dirichlet or Neumannconditions can be found by any means whatever, then that solution is the correctone, since only one exists. This result is the mathematical justification for themethod of images, which is discussed more fully in the next chapter.706