## Divergent integral

Let satisfy on . Show that the integral is convergent if and only if .

*PHD 6201*

**Proof: **Suppose there exists with . Apply the mean value property for the harmonic function and get

, where is the surface of the -dimensional unit ball. By Cauchy Schwarz inequality we get

, which means that

.

We finish by whe following inequality: . The inequality is true for any , and therefore, for we see that the integral is divergent. This means that the given integral is convergent if and only if .

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Categories: Analysis, Partial Differential Equations
harmonic, PDE

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