Problem set 3

1. Sum vs. product in Sobolev’s inequality. One proof of the Sobolev inequality first establishes by induction that \begin{equation}\tag{1} \| f \|_{n/(n-1)} \stackrel{\rm def}= \left(\int_{\R^n} |f(x)|^{n/(n-1)} dx\right)^{(n-1)/n} \leq C_n \prod_{k=1}^n \|D_kf\|_{L^1}^{1/n} \end{equation} and then uses the geometric/arithmetic mean inequality $(a_1a_2\cdots a_n)^{1/n} \leq (a_1+\cdots+a_n)/n$ to conclude \begin{equation}\tag{2} \| f \|_{n/(n-1)} \leq \frac{C_n}{n} \int_{\R^n} \sum_{k=1}^n |D_k f(x)|\,dx \end{equation} Show that (2) implies (1) (possibly with a different constant) by applying (2) to the function $\tilde f(x_1, \ldots, x_n) = f(\lambda_1 x_1, \ldots, \lambda_n x_n)$ for arbitrary $\lambda_k\geq0$ with $\lambda_1\lambda_2\dots\lambda_n = 1$.
2. About Schrödinger operators. A Schrödinger operator on some bounded domain $\Omega\subset\R^n$ is a linear elliptic operator of the form \[ \cH = -\Delta + V(x), \text{ i.e. }(\cH \phi)(x) = -\Delta\phi(x) + V(x)\phi(x), \] where the function $V:\Omega\to\R$ is called “the potential.” Schrödinger operators appear in quantum mechanics. The situation we consider here corresponds to the situation where the particle is constrained to move around in a box with shape $\Omega$. The operator $\cH$ is called the Hamiltonian for the particle. You can show that the operator $\cH$ has a discrete sequence of eigenvalues corresponding to a complete orthogonal set of eigenfunctions. In this problem you will derive a lower bound for the eigenvalues, under certain assumptions on the potential $V$. The interesting case here is where the potential $V$ may be negative, and may in fact be unbounded from below (e.g. the Coulomb potential in $\R^3$ is $V(x) = -C|x|^{-1}$).
   Assume $n\geq3$ and assume that the potential satisfies $V\in L^{n/2}(\Omega)$.
   1. Show that for any $\phi, \psi\in C_c^\infty(\Omega)$ one has \[ \left|\int_\Omega V(x)\phi(x)\psi(x) \,dx\right| \leq C \|V\|_{L^{n/2}}\,\|\phi\|_{H_0^1}\|\psi\|_{H_0^1} \]
   2. Show that the operator $\cH$ is bounded from $H_0^1(\Omega)$ to $H^{-1}(\Omega)$.
   3. Show that for any $\phi\in H^1_0(\Omega)$ one has \[ \left\langle \cH\phi, \phi\right\rangle = \int_\Omega \bigl\{ |\nabla\phi(x)|^2 + V(x)\phi(x)^2\bigr\}\, dx. \] A function $\phi\in H_0^1(\Omega)$ is an eigenfunction with eigenvalue $E$ if $\cH\phi = E\phi$. What does the above inequality imply for eigenfunctions?
   4. Show that for any $\phi\in C_c^\infty(\Omega)$ and any $h\in\R$ one has \[ \int_\Omega V(x)\phi(x)^2dx \geq -h \int_\Omega \phi^2dx - C\|V_h\|_{L^{n/2}} \|\phi\|_{H_0^1}^2 \] where $C\gt0$ does not depend on $\phi\in C_c^\infty(\Omega)$ or $h\in\R$, and where, by definition, \[ V_h(x) = \begin{cases} V(x)+h & V(x)\lt -h\\ 0 & V(x)\geq -h \end{cases} \]
   5. Show that for any given $V\in L^{n/2}(\Omega)$ there is a lower bound $-h_*\in\R$ for the operator $\cH$, i.e. for all $\phi\in C_c^\infty(\Omega)$ one has \[ \left\langle \cH\phi, \phi\right\rangle \geq -h_*\|\phi\|_{L^2}^2. \] (Hint: Use the previous parts of this problem, and in particular, show that$\|V_h\|_{L^{n/2}}\to0$ as $h\to\infty.$)
   6. Show that $-h_*$ from the previous problem is a lower bound for the eigenvalues of $\cH$, i.e. if $E$ is an eigenvalue of $\cH$, then $E\geq -h_*$.
   7. For which dimensions $n\geq3$ does the Coulomb potential \[ V(x) = - \frac{Q}{|x|^{n-2}} \] with $Q\gt0$ belong to $L^{n/2}(B_1(0))$?
3. About substitution operators. As always, $\Omega\subset\R^n$ is an open and bounded subset. Let $f:\R\to \R$ be a Lipschitz continuous function, with $|f(u)-f(v)|\leq M |u-v|$. Consider the substitution operator $\Phi$ which maps any given function $u:\Omega\to\R$ to the function $\Phi(u)$ defined by $\Phi(u)(x) = f(u(x))$. (So, $\Phi(u) = f\circ u$.)
   1. Show that for any $u\in L^p(\Omega)$ one has $\Phi u\in L^p(\Omega)$ where $p\in[1,\infty)$.
   2. Show that for any $u\in W^{1,p}(\Omega)$ one also has $\Phi u\in W^{1,p}(\Omega)$.
   3. Prove or disprove: for any two functions $u,v : \Omega\to\R$ one has \[ u, v\in L^p(\Omega) \implies \left\|\Phi u - \Phi v\right\|_{L^p} \leq C \|u-v\|_{L^p}, \] for some constant $C$ that does not depend on $u, v$.
   4. Prove or disprove: for any two functions $u,v : \Omega\to\R$ one has \[ u, v\in W^{1,p}(\Omega) \implies \left\|\Phi u - \Phi v\right\|_{W^{1,p}} \leq C \|u-v\|_{W^{1,p}}, \] for some constant $C$ that does not depend on $u, v$.