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2 votes
Accepted

Dense subspace of space of radial functions in $H^1_0(\Omega)$

Let me make your original approach rigorous. Wlog I'll assume that $\Omega=B(0,1)\subseteq \mathbb{R}^n$ (rescaling everything we can always reduce to this case). Let $u\in H_{0,\mathrm{rad}}^1(\Omega)...
Severin Schraven's user avatar
2 votes

Dense subspace of space of radial functions in $H^1_0(\Omega)$

In the other answer it is noted that one can redo the entire construction as in the non-radial case by choosing the mollifier and the cut-off function radially. Another approach is to directly utilize ...
Severin Schraven's user avatar
1 vote
Accepted

Integral function of bounded variation function derivative

Extend $f$ to all of $\mathbb{R}$ by setting $f(x) = f(a)$ for $x < a$ and $f(x) = f(b)$ for $x > b$. Then by the Fatou's lemma, \begin{align*} \int_{a}^{b} |f'(x)| \, \mathrm{d}x &=\int_{a}^...
Sangchul Lee's user avatar
1 vote

Dense subspace of space of radial functions in $H^1_0(\Omega)$

Let $u \in H^1_0(B)$ a radial function. For a given $\delta > 0$, consider $\varphi_\delta$ a standard radial mollifier. Now, let $\overline{u}$ the extension by zero of $u$ to $\mathbb{R}^N$ ...
Lucas Linhares's user avatar

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