Let $U \subseteq \mathbb{R}^n$ be an open connected set. Let $p>1$, and let $f \in W^{1,p}(U)$. I have recently heard that $f$ can be locally approximated in $W^{1,p}$ by polynomials. That is, there exist a sequence of polynomials $p_n$ such that $p_n \to f$ in $W^{1,p}_{loc}(U)$.

Furthermore, if $f$ is continuous, can we choose the $p_n$ to be uniformly convergent?

I would like to find a reference for such a proof. (Or if this is rather elementary, a proof sketch given here).

Actually, I don't really need a single sequence of polynomials $p_n$ which converges on all compact subsets. It suffices that for every arbitrarily small ball $B$ in $\Omega$, there would be a sequence $p_n^{B}$ that would converge to $f$. (That is, the sequence $p_n$ can depend on the ball).

I know it suffices to assume $f \in C^{\infty}$. Moreover, we can approximate uniformly $C^{\infty}$ maps, by polynomials, due to the Stone-Weierstrass theorem. But how can we approximate all the weak derivatives and the function simultaneously?

  • $\begingroup$ I am not sure this is correct, though I don't have a counterexample at hand. A different and more common "local approximation of $W^{1,p}$ function by polynomials" is approximation by piecewise polynomials. This is known as the "finite element method". $\endgroup$ – Kusma Aug 9 '18 at 9:15
  • $\begingroup$ There is a generalization of the Weierstrass approximation theorem that gives an approximation of any given (finite) order of derivatives on compact subsets (see for instance Sauvigny's PDE vol.1 book). After this we follow the scheme you propose. $\endgroup$ – Jose27 Aug 15 '18 at 4:41

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