In all our exercises in this sheet we will assume that \(\Omega\) is an open, bounded, and connected set with smooth boundary.
Exercise 1 (Lower bound for Dirichlet’s Energy). In this problem we will consider the Dirichlet Energy associated to the PDE \[\begin{equation} \label{eq:dirichlet} \left\{ \begin{array}{ll} -\Delta u =f& x\in \Omega\\ u=0 & x\in\partial \Omega \end{array} \right. \end{equation}\] \[E[u]=\frac{1}{2}\int_{\Omega}\left\lvert\nabla u(x)\right\rvert^2 dx - \int_{\Omega}f(x)u(x) dx.\]
Young’s inequality states that for any \(a,b\in\mathbb{R}\) and any Hölder conjugate numbers \(p,q\in (1,\infty)\) we have that \[\left\lvert ab\right\rvert \leq \frac{\left\lvert a\right\rvert^p}{p}+\frac{\left\lvert b\right\rvert^q}{q}.\] Consequently, for any \(\varepsilon>0\) we find that by replacing \(a\) with \(\left( p\varepsilon\right)^{\frac{1}{p}}a\) and \(b\) with \(\frac{b}{\left( p\varepsilon\right)^{\frac{1}{p}}}\) we get that \[\nonumber \left\lvert ab\right\rvert \leq \varepsilon\left\lvert a\right\rvert^p+\frac{\left\lvert b\right\rvert^q}{qp^{q-1} \varepsilon^{q-1}}\] and in particular that for any \(\varepsilon>0\) choosing \(p=q=2\) yields \[\nonumber %label{eq:specialised_young} \left\lvert ab\right\rvert \leq \varepsilon a^2 + \frac{b^2}{4\varepsilon}.\] Show that for any \(\varepsilon>0\) and any \[u\in V=\left\lbrace v\in C^1\left( \overline{\Omega}\right)\;|\; v=0\text{ on }\partial \Omega\right\rbrace\] we have that \[E[u] \geq \left( \frac{1}{2}- \varepsilon C_P\left( \Omega\right)^2\right)\left\lVert u\right\rVert_{H^1_0\left( \Omega\right)}^2 - \frac{1}{4\varepsilon}\left\lVert f\right\rVert^2_{L^2\left( \Omega\right)}\] where \(C_P\left( \Omega\right)\) is the Poincaré constant associated to the domain \(\Omega\).
Conclude that there exists a constant \(C>0\) such that \[\inf_{u\in V}E[u] \geq -C .\]
Exercise 2 (Uniqueness of weak solutions). Show that if \(u\) and \(v\) are weak solutions, in the sense defined in class, for \[\nonumber \left\{ \begin{array}{ll} -\Delta u =f& x\in \Omega\\ u=0 & x\in\partial \Omega \end{array} \right.\] then \(u=v\).
Exercise 3 (Uniqueness for a more general elliptic problem). Consider the linear, second-order, elliptic PDE \[\label{Q9} \begin{align} - \mathrm{div} (A \, \nabla u) + \textbf{b} \cdot \nabla u + c u = f \quad & \textrm{in } \Omega, \\ u = g \quad & \textrm{on } \partial \Omega, \end{align}\] where \(\Omega \subset \mathbb{R}^n\) is open and bounded with smooth boundary, \(A \in C^1(\overline{\Omega};\mathbb{R}^{n \times n})\), \(\textbf{b} \in C^1(\overline{\Omega};\mathbb{R}^n)\), and \(c,f,g \in C(\overline{\Omega})\). Assume that \(c\) is non-negatives, \(\mathrm{div}\textbf{b}=0\), and \(A\) is uniformly positive definite, i.e., there exists a constant \(\alpha>0\) such that \(y^T A(x) y \ge \alpha |y|^2\) for all \(y \in \mathbb{R}^n\), \(x \in \Omega\). Prove that \(\eqref{Q9}\) has at most one solution \(u \in C^2(\overline{\Omega})\).
Exercise 4. Consider the space \[V = \{ \varphi \in C^1(\overline{\Omega}) \, : \, \varphi=0 \textrm{ on } \partial \Omega, \; \varphi \ne 0\}\] and the functional \(E:V \to \mathbb{R}\) defined by \[E[v] = \frac{\displaystyle \int_\Omega |\nabla v(x)|^2 \, d x}{\displaystyle \int_\Omega |v(x)|^2 \, d x}.\] Suppose that \(u \in C^2(\overline{\Omega}) \cap V\) minimises \(E\) and show that \[\begin{aligned} - \Delta u & = \lambda u \quad \textrm{in } \Omega, \\ u & = 0 \quad \; \, \; \; \, \textrm{on } \partial \Omega, \end{aligned}\] where \(\lambda = E[u]\).