Let be harmonic ( ) and have continuous second partial derivatives on an open…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Partial differential equations and boundary-value problems · Partial differential equations and boundary-value problems

Conditions of Validity, Ch The three equations, their characteristics and what makes a problem well posed

Problem

Let uu be harmonic (∇2u=0\nabla^2u=0) and have continuous second partial derivatives on an open connected region V⊂R2V\subset\mathbb{R}^2, and let DR(r0)D_R(\mathbf{r}_0) be a closed disc of radius RR centred at r0∈V\mathbf{r}_0\in V, lying entirely inside VV. ((a)) Using Green's second identity in the plane, ∫A(u∇2ψ−ψ∇2u) dA=∮∂A(u ∂nψ−ψ ∂nu) ds,\int_A\bigl(u\nabla^2\psi-\psi\nabla^2u\bigr)\,\mathrm{d}A=\oint_{\partial A}\bigl(u\,\partial_n\psi-\psi\,\partial_n u\bigr)\,\mathrm{d}s, together with the two-dimensional fundamental solution ψ(r)=−ln⁡r\psi(r)=-\ln r (where r=∣r−r0∣r=|\mathbf{r}-\mathbf{r}_0|, and ∇2ψ=0\nabla^2\psi=0 for r≠0r\ne0), applied on the annulus A={ϵ≤r≤R}A=\{\epsilon\le r\le R\}, derive the mean-value property u(r0)=12πR∮r=Ru ds.u(\mathbf{r}_0)=\frac{1}{2\pi R}\oint_{r=R}u\,\mathrm{d}s. (Let ϵ→0+\epsilon\to0^+ at the end and use the continuity of uu at r0\mathbf{r}_0.) ((b)) Deduce the strong maximum principle: a non-constant harmonic function on a connected open region has no interior maximum or minimum. ((c)) Check the mean-value property for the harmonic temperature deviation u(x,y)=k xyu(x,y)=k\,xy with k=5.0 K m−2k=5.0\,\mathrm{K\,m^{-2}}, by evaluating its average directly over the circle x2+y2=a2x^2+y^2=a^2 with a=0.20 ma=0.20\,\mathrm{m} centred at the origin, and comparing with u(0,0)u(0,0).

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