# Sobolev inequality with holes

Classical Sobolev inequality says, $$n\geq 3$$, we have $$$$\left(\int_{\mathbb{R}^n}|u|^{2 n /(n-2)}\right)^{(n-2) /(2 n) } \leq C(n)\left(\int_{\mathbb{R}^n}|\nabla u|^{2}\right)^{1 / 2}\quad \text{for all}\,u\in D^{1,2}(\mathbb{R}^n).$$$$ for some constant $$C(n)$$ depends on $$n$$. Here the space $$D^{1,2}(\mathbb{R}^n)=\{u\in L^{2}(\mathbb{R}^n):|\nabla u|\in L^{2 n /(n-2)}(\mathbb{R}^n)\}$$.

My question is whether it is ture we have the Sobolev type inequality $$$$\left(\int_{\Omega}|u|^{2 n /(n-2)}\right)^{(n-2) /(2 n) } \leq C_{1}(n,\Omega)\left(\int_{\Omega}|\nabla u|^{2}\right)^{1 / 2},$$$$ with $$\Omega=\mathbb{R}^n\backslash B_1(0)$$, $$u\in E=\{u \in L^{2n /(n-2)}(\Omega):|\nabla u|\in L^{2}(\Omega)\}$$, $$C_{1}(n,\Omega)$$ is a constant only depends on $$n$$ and $$\Omega$$.

• It is useful to recall the definition of the space $D^{1,2}$ here.
– Medo
Apr 25, 2023 at 15:39
• In the specific case: since $B_1(0)$ is convex, you can copy over Nirenberg's 1959 proof (the one integrating over rectangles) with almost zero change. Apr 26, 2023 at 1:16
• @WillieWong Could you be more precise: which paper and which result are you referring to? Apr 26, 2023 at 6:18
• @giorgio-metafune I believe he refers to Lecture II of Louis Nirenberg. "On elliptic partial differential equations." Annali della Scuola Normale Superiore di Pisa-Scienze Fisiche e Matematiche 13, no. 2 (1959): 115-162.
– cs89
Apr 26, 2023 at 7:54
• There is also a recent rewriting Alberto Fiorenza, Maria Rosaria Formica, Tomáš G. Roskovec, and Filip Soudský. "Detailed proof of classical Gagliardo–Nirenberg interpolation inequality with historical remarks" Zeitschrift für Analysis und ihre Anwendungen 40, no. 2 (2021): 217-236 which is very helpful.
– cs89
Apr 26, 2023 at 7:55

This paper focuses on the case where no vanishing condition is imposed on $$\partial\Omega$$.
The case you mention follows from Theorem 2.1 with $$m=1$$, $$p=2$$, $$k=0$$, $$r = \frac{2n}{n-2}$$. Since $$n\geq 3$$, beware that you are in the "particular case" ($$k = 0$$ and $$m p < n$$) which requires the additionnal assumption that $$w$$ tends to 0 at infinity, or that $$w \in L^{q'}$$ for some $$q'<\infty$$ (which is your case with $$q' = 2$$).