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An integral identity for Harmonic Functions

Partial Differential Equations
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An integral identity for Harmonic Functions


Post by r9m » Sat May 20, 2017 11:07 am

If $u$ be a Harmonic Function in a open connected set $\Omega \subset \mathbb{R}^n$ and $\overline{B(x_0,R)} \subset \Omega$ (the closed ball of radius $R$ centered at $x_0 \in \Omega$).

(i) Show that: $$\int_{\partial B(0,1)} u(x_0 + ry)u(x_0 + Ry)\,d\sigma(y) = \int_{\partial B(0,1)} u^2(x_0 + cy)\,d\sigma(y)$$ where, $r \le c \le R$ and $c^2 = rR$.

(ii) Using the above identity show that if $u$ is locally constant, then it is constant in $\Omega$.

[$\partial B(0,1) \equiv S^{n-1}$ is the boundary of the unit ball in $\mathbb{R}^n$ and $\,d\sigma(y)$ is the surface measure on $\partial B(0,1)$].
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