Elliptic boundary problem. PDE

Question

I am working on several PDEs problems for training the knolowdege aquired during the course I took at University and I get stacked in the following one. I understand all the terms and concepts involved in the following questions but I actually don't know how to solve them correctly. I would really appreciate if anyone could help me solving this problem or even knows if it is already solved in any book or reference. Thank you in advance.

Let $\Omega\subset\mathbb{R}^N$ be a bounded regular neihgborhood and lets consider the following elliptic problem: $$-\Delta u+u=0\text{ in }\Omega$$ $$\frac{\partial u}{\partial n}=g(x)\text{ in }\Gamma=\partial\Omega$$ where $f\in L^{2}(\Omega)$ and $g\in L^2(\partial\Omega)$. Lets consider now the bilinear form defined as follows: $$a:H^1(\Omega)\times H^1(\Omega)\rightarrow\mathbb{R}$$ $$a(u,v)=\int_{\Omega}\nabla u\nabla v +uv $$

I want to prove the following statements:

1) The map $L:H^1(\Omega)\rightarrow\mathbb{R}$ given by $L(v)=\int_{\Omega} fv dx + \int_{\partial \Omega} gvd\sigma$ is linear and continuous.

2) The weak formulation of the elliptic problem is the following one: find $u\in H^1(\Omega)$ satisfying that $a(u,v)=L(v)$ for all $v\in H^1(\Omega)$.

3)Formulate and prove an existence and uniqueness theorem for weak solutions.

Answer 1

$$ \begin{cases} -\Delta u+u=f&\text{ in }\Omega \\ \frac{\partial u}{\partial n}=g(x)&\text{ on }\Gamma=\partial\Omega \end{cases}$$ where $\Omega^{\text{bdd}}\subset \mathbb{R}^n, f\in L^{2}(\Omega), g\in L^2(\partial\Omega)$.

• 1) The map $L:H^1(\Omega)\rightarrow\mathbb{R}$ given by $L(v)=\int_{\Omega} fv\ \text{d}x + \int_{\partial \Omega} g\ \gamma v\ \text{d}\sigma$ is linear and continuous.

(Notice that I applied the linear continuous trace operator $\gamma:H^1(\Omega) \to L^2(\partial \Omega)$ in the boundary integral to make it well defined) Linearity of $L$ is clear. For continuity we show boundedness (for a linear map continuity and boundedness are equivalent), i.e. $|L(v)|\leq C ||v||_{H^1(\Omega)}$ for all $v \in H^1(\Omega)$. $$|L(v)|\leq ||f||_{L^2(\Omega)} \underbrace{||v||_{L^2(\Omega)}}_{\leq||v||_{H^1(\Omega)}}+||g||_{L^2(\partial \Omega)} \underbrace{||\gamma v||_{L^2(\partial\Omega)}}_{\leq ||\gamma||\ ||v||_{H^1(\Omega)}}\leq C \ ||v||_{H^1(\Omega)} $$

• 2) The weak formulation of the elliptic problem is the following one: find $u\in H^1(\Omega)$ satisfying that $a(u,v)=L(v)$ for all $v\in H^1(\Omega)$.

Just apply some test function $v\in H^1(\Omega)$ to your PDE and do integration by parts to get $$\underbrace{\int_\Omega \nabla u \nabla v +u v \ \text{d}x}_{=:a(u,v)}= \underbrace{\int_\Omega f v \ \text{d}x+\int_{\partial \Omega} g \ \gamma v \ \text{d}\sigma}_{=:L(v)}$$

• 3)Formulate and prove an existence and uniqueness theorem for weak solutions.

Look up the theorem by Lax-Milgram (for a proof see for example 'Partial Differential Equations' by Evans). It tells you the following:

Let $a$ be a continuous, elliptic bilinear form on a Hilbert space $V$ and $L \in V'$. Then there exists a unique $u \in V$ to the variational problem $a(u,v)=L(v)$ for all $v \in V$.

So it only remains to check if $a$ is continuous and elliptic. This is indeed true since

$$\begin{align}&|a(u,v)| \leq ||\nabla u||_{L^2(\Omega)} ||\nabla v||_{L^2(\Omega)}+||u||_{L^2(\Omega)} ||v||_{L^2(\Omega)}\leq 2 \ ||u||_{H^1(\Omega)}||v||_{H^1(\Omega)} \\ &a(u,u) =\int_\Omega |\nabla u|^2 +|u|^2 \ \text{d}x=||v||^2_{H^1(\Omega)}\end{align} $$