Paper 3, Section II, B

Vector Calculus | Part IA, 2017

Let SS be a piecewise smooth closed surface in R3\mathbb{R}^{3} which is the boundary of a volume VV.

(a) The smooth functions ϕ\phi and ϕ1\phi_{1} defined on R3\mathbb{R}^{3} satisfy

2ϕ=2ϕ1=0\nabla^{2} \phi=\nabla^{2} \phi_{1}=0

in VV and ϕ(x)=ϕ1(x)=f(x)\phi(\mathbf{x})=\phi_{1}(\mathbf{x})=f(\mathbf{x}) on SS. By considering an integral of ψψ\boldsymbol{\nabla} \psi \cdot \boldsymbol{\nabla} \psi, where ψ=ϕϕ1\psi=\phi-\phi_{1}, show that ϕ1=ϕ\phi_{1}=\phi.

(b) The smooth function uu defined on R3\mathbb{R}^{3} satisfies u(x)=f(x)+Cu(\mathbf{x})=f(\mathbf{x})+C on SS, where ff is the function in part (a) and CC is constant. Show that

VuudVVϕϕdV\int_{V} \nabla u \cdot \nabla u d V \geqslant \int_{V} \nabla \phi \cdot \nabla \phi d V

where ϕ\phi is the function in part (a). When does equality hold?

(c) The smooth function w(x,t)w(\mathbf{x}, t) satisfies

2w=wt\nabla^{2} w=\frac{\partial w}{\partial t}

in VV and wt=0\frac{\partial w}{\partial t}=0 on SS for all tt. Show that

ddtVwwdV0\frac{d}{d t} \int_{V} \nabla w \cdot \nabla w d V \leqslant 0

with equality only if 2w=0\nabla^{2} w=0 in VV.

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