## Green representation and a special representation

Suppose , where is a radial function and has compact support in , . Prove that

Compare with , where .

Since has compact support, by Green’s representation

So is a radial function. We know that

By the divergence theorem

here is a centered ball with radius and positive orientation.

By the Green’s second identity,

So

Almut Burchard. A short course on Rearrangement Inequalities.

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