let $\Omega$ be a domain in $R^n$ and let $\delta(x)=dist(x,\partial \Omega)$ and assume that $u_k \to 0$ weakly in$ W_{1,p}^\circ$ (sobolev space with trace =zero).

I can conclude that $u_k\to0$ in $L_p^{loc}(\Omega)$.

my question is can I conclude that $ \int_{\Omega '}\left |\frac{u_k}{\delta(x)} \right |^p \to 0$ as $k\to \infty$ for every subset $\Omega '\subset \Omega$ , $dist(\Omega',\partial \Omega)>0 $?

my attamp :

I took a monotone increasing compact sets $\Omega_m\to\Omega '$

I know that $lim_{m\to \infty}lim_{k\to \infty} \int_{\Omega_m }\left |\frac{u_k}{\delta(x)} \right |^p = 0$

but how can I deduce that the latter limit converge to $ lim _{k\to \infty}\int_{\Omega '}\left |\frac{u_k}{\delta(x)} \right |^p $ as $m\to \infty$ ?

thank you


got the answer

the assertion is true provided that the boundary is compact.

we examine the limit $ lim _{k \to \infty }\int_{\Omega '}\left | \frac{u_k}{\delta} \right |^p$

weak convergence imply that $\int_{\Omega '}\left | \frac{u}{\delta} \right |^p<C$ .

we take large ball with radius R and examine the integral in $B\cap \Omega'$

from the convergence to zero in $L_{p}^{loc} $ we also know that the latter converge to zero

we left with integral outside the ball which is estimated as $\frac{1}{R^p}\int_{\Omega '}\left | \frac{u}{\delta} \right |^p<\frac{C}{R^p}$

take $R\to \infty $ and get that this integral is also zero


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