Uniqueness/existence of weak solution

Provide a weak formulation for the following boundary value problem $$ -\Delta u=f, \quad \text { in } \Omega, \quad \text { and }\left.\quad u\right|_{\partial \Omega}=0 \space \space \space \space \space \space (1) $$ Prove that for any $f \in L^{2}(\Omega)$ there exists a unique weak solution of (1).

My attempt:

$$ \|\nabla u\|_{L^{2}(\Omega)}^{2}=\int \nabla u \cdot \nabla u=\int f u \leq\|f\|_{L^{2}(\Omega)}\|u\|_{L^{2}(\Omega)} \leq C_{\epsilon}\|f\|_{L^{2}(\Omega)}^{2}+\epsilon\|u\|_{L^{2}(\Omega)}^{2} . $$ Using the Poincare inequality for $u:\|u\|_{L^{2}(\Omega)} \leq C\|\nabla u\|_{L^{2}(\Omega)}$ and taking $\epsilon$ sufficiently small we have $$ \|\nabla u\|_{L^{2}(\Omega)} \leq C\|f\|_{L^{2}(\Omega)} . $$ So, assuming there are two weak solutions $u, v \in H_{0}^{1}(\Omega)$, consider $w=u-v$ which solves $\Delta w=0$ with $w=0$ on $\partial \Omega$. Our above estimate establishes that $w \equiv 0$ and so $u=v$.

I know this proves uniqueness, but how can I prove existence?

Can anyone show how I can apply Lax-Milgram here?

Solution 1:

The weak form of your equation is given by $$-\int_\Omega \nabla \varphi \cdot \nabla u=\int_\Omega f\varphi ,\quad \varphi \in H_0^1(\Omega )$$ If you set $$B(u,\varphi ):=-\int_\Omega \nabla \varphi \cdot \nabla u\quad \text{and}\quad L(\varphi ):=\int_\Omega f\varphi. $$ The existence of a weak solution of your problem is equivalent than proving the existence of $u\in H_0^1(\Omega )$ s.t. $$B(u,\varphi )=L(\varphi ),$$ for all $\varphi \in H_0^1(\Omega )$.

Notice that by Riesz-representation Theorem, there is a unique $\hat f\in H_0^1(\Omega )$ s.t. $$L(\varphi )=\left<\hat f,\varphi \right>_{H_0^1(\Omega )},$$ for all $\varphi \in H_0^1(\Omega )$. Now, existence of $u\in H_0^1(\Omega )$ s.t. $$B(u,\varphi )=\left<\hat f,\varphi \right>_{H_0^1(\Omega )},$$ for all $\varphi \in H_0^1(\Omega )$ follows from Lax-Milgram Theorem.

What remains to prove to be able to use Lax-Milgram Theorem is that $L$ is linear and continuous, and that $B$ is bilinear, continuous and coercive, i.e. that there is $C>0$ s.t. $|B(v,v)|\geq C\|v\|_{H_0^1(\Omega )}$ for all $v\in H_0^1(\Omega )$.