I want to show that

$\displaystyle \frac{1}{2\pi r}\int_{\partial B(x,r)}u(y)\,\mathrm{d}s(y)=\frac{1}{|S^1|}\int_{S^1}u(x+r\theta)\,\mathrm{d}s(\theta)$

This is essentially a shift and dilation from (or) to the unit sphere.

I defined a diffeomorphism

$\Phi\colon S^1\rightarrow\partial B(x,r)\subset\mathbb{R}^2\\ \theta\mapsto x+r\theta$

where $\theta$ is a point on $S^1$.

It follows that $|\det(D\Phi(S^1))|= r$

and therefore

$\displaystyle \frac{1}{2\pi r}\int_{\partial B(x,r)}u(y)\,\mathrm{d}s(y)=\frac{1}{2\pi r}\int_{S^1}u(\Phi(S^1))r\,\mathrm{d}s(\theta)=\frac{1}{2\pi(=|S^1|)}\int_{S^1}u(x+r\theta)\,\mathrm{d}s(\theta)$

Is this correct? If yes, I appreciate to have a look at solutions using a different approach.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.