Prove boundedness for this bilinear form

Question

I have the following bilinear form

$$a(u,v) = \int_0^1 \bigg(\frac{du}{dx}\frac{dv}{dx}+v\frac{du}{dx}+uv \bigg) \, dx$$

defined on $H^1_0(0,1)$ where $u(0)=v(0)=u(1)=v(1)=0$. How do I show that this bilinear form is bounded?

Unfortunately, this bilinear form is not symmetric so I can't apply Cauchy-Schwarz directly.

The norm on $H_0^1(0,1)$ is

$$\lVert v \rVert_{H_0^1(0,1)} = \int_0^1\bigg(\Big(\frac{dv}{dx}\Big)^2+v^2 \bigg) \, dx$$

and we see that

\begin{align} a(v,v) = & \int_0^1\bigg(\Big(\frac{dv}{dx}\Big)^2+v\frac{dv}{dx} + v^2 \bigg) \, dx \\ = & \int_0^1\bigg(\Big(\frac{dv}{dx}\Big)^2+\frac 12 \frac{d}{dx}\big(v^2\big) + v^2 \bigg) \, dx \\ = & \int_0^1\bigg(\Big(\frac{dv}{dx}\Big)^2+v^2 \bigg) \, dx + \bigg[\frac 12 v^2 \bigg]_0^1\\ = & \int_0^1\bigg(\Big(\frac{dv}{dx}\Big)^2+v^2 \bigg) \, dx \\ = & \lVert v \rVert_{H_0^1(0,1)} \end{align}

But I couldn't make much progress after this.

Any hints?

Thanks!

Answer 1

Cauchy-Schwarz works fine. If you apply it, you get that \begin{align*} |a(u,v)|&\leq \|v'\|_{L^2(0,1)}\|u'\|_{L^2(0,1)}+\|u'\|_{L^2(0,1)}\|v\|_{L^2(0,1)}+\|u\|_{L^2(0,1)}\|v\|_{L^2(0,1)}\\ &\leq C\|u\|_{H^1_0(0,1)}\|v\|_{H^1_0(0,1)}, \end{align*} since $\|f\|_{H^1_0(0,1)}=\left(\|f\|_{L^2(0,1)}^2+\|f'\|_{L^2(0,1)}^2\right)^{1/2}\geq \frac{\sqrt{2}}{2}\left(\|f\|_{L^2(0,1)}+\|f'\|_{L^2(0,1)}\right).$