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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.

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    Starting with your statement (1), one needs to show (a) linearity of the function, and (b) continuity, which is equivalent to boundedness given (a) linearity. That is, you want to show there exists a constant $C\gt 0$ such that $\forall v\in H^1(\Omega) \; |L(v)| \le C ||v||_1$, where the latter denotes the $H^1$-norm of $v$. – 2017-02-03
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    Please do not edit your Question to remove its substantive content. I have rolled back your recent edits to the title and body of the Question in order to restore the important information. – 2017-02-04

1 Answers 1

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$$ \begin{cases} -\Delta u+u=f&\text{ in }\Omega \\ \frac{\partial u}{\partial n}=g(x)&\text{ on }\Gamma=\partial\Omega \end{cases}$$ 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

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