Let $\Omega= B_1(0)$ the unit ball in $\mathbb{R^2}$. How to prove that for $0<k<1/2$ the function $v(x)= |\log(\|x\|)|^k$ is in
$$H^1(\Omega)=\left\{ f\in L^2(\Omega) : \frac{\partial f}{\partial x_i} \in L^2(\Omega) , i=1,2\right\}?$$
Asked
Active
Viewed 38 times
1
-
Welcome to stackexchange. What have you tried so far? – Mars Plastic Mar 13 '19 at 16:17
-
First step first: did you show that $v \in L^2(\Omega)$? – Umberto P. Mar 13 '19 at 17:48