Continuous bilinear form on $H_{0}^{1}(-1,1)$

Question

Consider the bilinear form defined on $H_{0}^{1}(-1,1)$ by $$a(u,v)=\int_{-1}^{1} (u'v' +uv-\lambda.u(0).v)$$

Where $|\lambda |<\sqrt{2}$ is fixed.

Prove that a is continuous and coercive.

I have tried to estimate $|u (0)|$, that is, I have to find a positive number M s.t $ |u(0)| \le M. \left\| u \right\|_{H_{0}^{1}}$ but i stuck here.

Thanks everyone !!

Answer 1

In dimension $1$ you have the embedding $H^1_0(\Omega) \subset C^0(\Omega)$. Without using Sobolev embeddings you can prove the result by a simple calculation: $$|u(0)|^2=\left|\int_{-1}^0 u'(x) dx \right|^2 \leq \int_{-1}^0 |u'(x)|^2 dx \int_{-1}^0 1^2 dx $$ so: $$|u(0)| \leq \sqrt{\int_{-1}^{0} |u'(x)|^2 }\leq \Vert u \Vert_{H^1_0} $$