# Application of Poincaré inequality on $W_0^{1,2}(\Omega)$

Let $$\Omega$$ be a bounded Lipschitz domain and let $$\mathcal{S}=\{u\in W^{1,2}(\Omega)\colon\int_\Omega u^3dx=0\}$$ Then there exists some constant $$C(\Omega)>0$$ such that $$\int_\Omega|u|^2dx\leq C\int_\Omega \|\nabla u\|^2dx.$$ This already really looks likes Poincaré inequality, besides the fact that $$u\in\mathcal{S}$$ does not need to have non-zero trace. Or is this non-zero trace property inherited by the fact that the $$\int_\Omega u^3dx$$ vanishes?

• What is the dimension of $\Omega$? Is $\Omega$ connected? – gerw Feb 13 at 7:54
• $\Omega\subset\mathbb{R}^3$ is a bounded Lipschitz domain. – Lucas Smits Feb 13 at 7:56

The inequality is indeed a Poincare inequality, but not the classical one for functions that vanish on the boundary. When $$\Omega$$ is a bounded Lipschitz domain, Poincare's inequality holds for any subspace $$S:=\{u\in W^{1,2}(\Omega)\ |\ G(u)=0 \}$$ where $$G:W^{1,2}(\Omega)\rightarrow\mathbb{R}$$ is weakly continuous and has the property $$u=const\ \wedge\ G(u) =0\quad\Rightarrow\quad u=0.$$ Here is proof by contradiction: Assume Poincare's inequality does not hold for all $$u\in S$$. Then there is a sequence $$(u_n)\subset S$$ such that $$||u_n||_2 \geq n ||\nabla u_n||_2.$$ Put $$v_n:=\frac{u_n}{||u_n||_2}$$. Then $$(v_n)$$ is a bounded sequence in $$W^{1,2}(\Omega)$$, and you can therefore find a weakly convergent subsequence (still denoted by $$(v_n)$$) with $$v_n \rightharpoonup v$$ for some $$v\in W^{1,2}(\Omega)$$. Since $$G$$ is weakly continuous, we even have $$G(v)=0$$. By the compact embedding $$W^{1,2}(\Omega)\hookrightarrow L^2(\Omega)$$, we further deduce $$||v||_2=1$$. Since $$||\nabla v_n||_2\rightarrow 0$$, we also have $$\nabla v$$=0. But this means that $$v=const$$, whence the property of $$G$$ implies v=0. This contradicts $$||v||_2=1$$.
In your case, you can put $$G(u):=\int_\Omega u^3 dx$$. The weak continuity of G then follows from the compact embedding $$W^{1,2}(\Omega)\hookrightarrow L^3(\Omega)$$, which holds when $$\Omega\subset\mathbb{R}^3$$.