The inclusion of Sobolev spaces

Question

Let $\Omega$ denote a bounded open set in $\mathbb R^n$ with sufficiently smooth boundary. If $u\in H_m^0(\Omega ) \cap H_k^{loc}(\Omega )$ with $m,k$ two positive integers, then can we conclude that $u \in H_k(\Omega )$?

Answer 1

Consider the function $f:[-1,1]\rightarrow\mathbb{R}$ defined by $$f(x)=\sqrt{1-x^2}$$

Note that $f\in H_1^0([-1,1])\cap H_2^{loc}([-1,1])$, but $f\notin H_2([-1,1])$.