I’m considering a $H^1$ function u on a open domain D. Is the integral:


$$ \int\_{\partial B\_r(x)} u \hspace{2pt}dH^{n-1}$$


continuous with respect to x?


I tried to prove that it’s differential by showing that the derivative can be written as the integration:


$$ \int\_{\partial B\_r(x)} Du \hspace{2pt}dH^{n-1}$$


But it’s just right a.e. From which we can deduce that it’s continuous a.e.. Is there any possibility to show that it’s continuous everywhere?



About the continuity of the integral on the boundary of a ball
I would say so. Denote your integral by $b\_u(x)=\int\_{|x-y|=r}u(y)dH^{n-1}$.
Approximate $u$ in $H^1$ with test functions $u\_j$. The property is certainly true for $u\_j$ thus it is enough to prove that $b\_{u\_j}\to b\_u$ uniformly. You can estimate $|b\_{u\_j}(x)-b\_u(x)|$ with the $L^2$ norm of the trace of $u-u\_j$ at $\partial B\_r(x)$, which is controlled by the $H^1$ norm of $u\_j-u$ (for fixed $r$)


