Math 6410 • ODE

Let \(G\) be an open set in \(\mathbb {R}^n\) with \[ \sup \{\lvert x_1\rvert :(x_1,x_2,\ldots ,x_n)\in G\}=K<+\infty . \] Then \[ \lVert \varphi \rVert _{L^2(G)}\le 2K\lVert \partial _1\varphi \rVert _{L^2(G)}, \quad \forall \varphi \in H_0^1(G). \tag {6} \] \[ \left (\text {Hint: For }\varphi \in C_0^\infty (G)\text { compute } \frac {\partial }{\partial x_1}(x_1\lvert \varphi (x)\rvert ^2)\right . \] \[ \left .\text {and use the divergence theorem.}\right ) \] Use (6) to obtain Poincaré’s inequality for functions \(\varphi \in H_0^1(G)\), whenever \(G\) is a bounded domain.