Partial Differential Equations III & V
Problem Class 5

Exercise 1 (Hölder inequality). The so-called Hölder inequality which states that for for (measurable) set \(E\subseteq \mathbb{R}^n\) and any \(f\in L^p\left( E\right)\) and \(g\in L^q\left( E\right)\) where \(p\) and \(q\) Hölder conjugate, i.e. \(p,q\in [1,\infty]\) and \(1/p+1/q=1\) we have that¹ \[\begin{equation} \nonumber %\label{eq:holder} \int_{E}\left\lvert f(x)g(x)\right\rvert dx \leq \left\lVert f\right\rVert_{L^p\left( E\right)}\left\lVert g\right\rVert_{L^q\left( E\right)}. \end{equation}\] where \(L^p\left( E\right)\) is defined like \(L^p\left( \mathbb{R}^n\right)\) where the set on which we integrate is \(E\) instead of \(\mathbb{R}^n\).

1. Prove Hölder inequality when \(E=\mathbb{R}^n\), \(p=\infty\) and \(q=1\). You may assume that \(f\) is bounded on \(\mathbb{R}^n\) (and as such use \(\sup\) instead of \(\mathrm{esssup}\)).

2. Show that if \(f\in C_c\left( \mathbb{R}^n\right)\) and \(g\in L^p\left( \mathbb{R}^n\right)\) then there exists a compact set \(K\subset \mathbb{R}^n\) such that \[\int_{\mathbb{R}^n} \left\lvert f(x)g(x)\right\rvert dx \leq \left\lVert f\right\rVert_{L^\infty}|K|^{\frac{1}{q}} \left( \int_{K}\left\lvert g(x)\right\rvert^pdx\right)^{\frac{1}{p}}.\]

3. Use Hölder inequality to show that if \(f\in L^p\left( \mathbb{R}^n\right)\) and \(g\in L^q\left( \mathbb{R}^n\right)\) with \(p\) and \(q\) Hölder conjugates we have that \[\sup_{x\in \mathbb{R}^n}\left\lvert f\ast g (x)\right\rvert=\left\lVert f\ast g\right\rVert_{L^\infty\left( \mathbb{R}^n\right)}\leq \left\lVert f\right\rVert_{L^p\left( \mathbb{R}^n\right)} \left\lVert g\right\rVert_{L^q\left( \mathbb{R}^n\right)}.\]

Exercise 2 (Poincaré-like inequalities).

1. Using the formula \[\begin{equation} \nonumber %\label{eq:FTC} f(x)-f(y)=\int_{y}^x f^\prime (s)ds, \end{equation}\] which holds for any \(C^1\) function on an interval that contains \(x\) and \(y\), together with Hölder’s inequality show that for any Hölder conjugates \(p\) and \(q\), any \(f\in C^1\left( [a,b]\right)\), and any \(c\in [a,b]\) we have that \[\begin{equation} \nonumber %\label{eq:poincare_general_infty} \sup_{x\in [a,b]}\left\lvert f(x)-f(c)\right\rvert \leq \left( b-a\right)^{\frac{1}{q}}\left\lVert f^\prime\right\rVert_{L^p\left( [a,b]\right)}, \end{equation}\] where \(q\) is the Hölder conjugate of \(p\).

2. Conclude that for any \(p\in [1,\infty]\) and \(r\in [1,\infty)\), any \(f\in C^1\left( [a,b]\right)\), and any \(c\in [a,b]\) we have that \[\begin{equation} \nonumber %\label{eq:poincare_general_I} \left( \int_{a}^b\left\lvert f(x)-f(c)\right\rvert^r dx\right)^{\frac{1}{r}} \leq \left( b-a\right)^{\frac{1}{q}+\frac{1}{r}}\left\lVert f^\prime\right\rVert_{L^p\left( [a,b]\right)}. \end{equation}\] or equivalently \[\begin{equation} \nonumber %\label{eq:poincare_general_II} \left\lVert f-f(c)\right\rVert_{L^r\left( [a,b]\right)}\leq \left( b-a\right)^{\frac{1}{q}+\frac{1}{r}}\left\lVert f^\prime\right\rVert_{L^p\left( [a,b]\right)}. \end{equation}\] Note that due to our previous sub-question the above remains true when \(r=\infty\).

3. Use sub-question [item:basic_ineq] to conclude that for any \(f\in C^1\left( [a,b]\right)\) and any \(p\in[1,\infty]\) we have \[\begin{equation} \nonumber %\label{eq:poincare_average} \left\lVert f-\overline{f}\right\rVert_{L^p\left( [a,b]\right)} \leq \left( b-a\right)\left\lVert f^\prime\right\rVert_{L^p\left( [a,b]\right)} \end{equation}\] where \[\overline{f} = \frac{1}{b-a}\int_a^b f(s)ds.\]

4. Use sub-question [item:basic_ineq] to conclude that for any \(f\in C^1\left( [a,b]\right)\) such that \(f(c)=0\) for some \(c\in [a,b]\) and any \(p\in[1,\infty]\) we have \[\begin{equation} \nonumber %\label{eq:poincare_dirichlet} \left\lVert f\right\rVert_{L^p\left( [a,b]\right)} \leq \left( b-a\right)\left\lVert f^\prime\right\rVert_{L^p\left( [a,b]\right)}. \end{equation}\]

Exercise 3 (The Poincaré constant).

1. Let \(\Omega\) be an open bounded domain in \(\mathbb{R}^n\) with smooth boundary. Consider the PDE \[\label{EVP} \begin{align} - \Delta u(x) & = \lambda u(x) \quad \textrm{in } \Omega, \\ u(x) & = 0 \quad \; \, \; \textrm{on } \partial \Omega, \end{align}\] where \(u\in C^2\left( \overline{\Omega}\right)\) and \(\lambda \in \mathbb{R}\). By multiplying the above by \(u\) and integrating by parts show that if \(u\ne 0\) we have that \[\begin{equation} \label{Rayleigh} \lambda = \frac{\displaystyle \int_\Omega |\nabla u(x)|^2 \, d x}{\displaystyle \int_\Omega u^2(x) \, d x} \geq 0. \end{equation}\] This is known as the Rayleigh quotient formula for the eigenvalue \(\lambda\) in terms of the eigenfunction \(u\).
   Remark: This procedure, where we multiplied by a function, integrated, and recovered a functional that helps us understand our equation better is known as the energy method (the functional we found is our “energy”). This is an important method which will repeat in this module.
   Remark: We could allow \(u\) to have values in \(\mathbb{C}\). In that case we would multiply our PDE with \(\overline{u}\) and conclude that \[\lambda = \frac{\displaystyle \int_\Omega |\nabla u(x)|^2 \, d x}{\displaystyle \int_\Omega \left\lvert u(x)\right\rvert^2 \, d x} \geq 0.\]

2. Show that any \(\lambda\) that satisfies our PDE must satisfy \[\lambda \geq \frac{1}{C_P^2}\] where \(C_P\) is the Poincaré inequality that is associated to \(\Omega\) and the Dirichlet boundary condition, i.e. the best constant for which \[\int_{\Omega}\left\lvert u(x)\right\rvert^2 dx \leq C_P^2 \int_{\Omega}\left\lvert\nabla u(x)\right\rvert^2dx.\] Remark: The above implies that \[\min_{\textrm{eigenvalues}}\lambda \geq \frac{1}{C_P^2}.\] One can in fact show that there is equality in the above. This is connected to minimising the energy \[E[u] = \frac{\displaystyle \int_\Omega |\nabla u(x)|^2 \, d x}{\displaystyle \int_\Omega |v(x)|^2 \, d x}\] over an appropriate space of functions.