Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 344 / 2 / d

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- Solution d

Solution

For

h = η sin (q x)

, the linearized mean curvature is

K = (\partial_{x}^{2} + \partial_{y}^{2}) h = - q^{2} η sin (q x) .

The boundary value in the upper phase is therefore

g (x, 0) = \frac{σ q ^{2} η}{2 α ϕ _{B}} sin (q x) .

The decaying harmonic extension into

z > 0

is

g (x, z) = \frac{σ q ^{2} η}{2 α ϕ _{B}} e^{- ∣ q ∣ z} sin (q x) .

With the normal directed into the upper half-space,

J_{n} (x, 0^{+}) = - M α \partial_{z} g (x, 0^{+}) = \frac{M σ ∣ q ∣ ^{3}}{2 ϕ _{B}} η sin (q x) .

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