On the minimisation of the Peak-to-average ratio

Abstract

Let $\Omega \Subset \mathbb R^n$ and a continuous function $\mathrm H$ be given, where $n,k,N \in \mathbb N$. For $p\in [1,\infty]$, we consider the functional \[ \mathrm E_p(u) := \big\| \mathrm H \big(\cdot,u,\mathrm D u, \ldots, \mathrm D^ku \big) \big\|_{\mathrm L^p(\Omega)},\ \ \ u\in \mathrm W^{k,p}(\Omega;\mathbb R^N). \] We are interested in the $L^\infty$ variational problem \[ \mathrm C_{\infty,p}(u_\infty)\, =\, \inf \Big\{\mathrm C_{\infty,p}(u) \ : \ u\in \mathrm W^{k,\infty}_\varphi(\Omega;\mathbb R^N), \ \mathrm E_1(u)\neq 0 \Big\}, \] where $\varphi\in \mathrm W^{k,\infty}(\Omega;\mathbb R^N)$, $p$ is fixed, and \[ \mathrm C_{\infty,p}(u)\, := \, \frac{\mathrm E_\infty(u)}{\mathrm E_p(u)} . \] The variational problem is ill-posed. $\mathrm C_{\infty,2}$ is known as the ``Crest factor" and arises as the ``peak--to--average ratio" problem in various applications, including eg. nuclear reactors and signal processing in sound engineering. We solve it by characterising the set of minimisers as the set of strong solutions to the eigenvalue Dirichlet problem for the fully nonlinear PDE \[ \left\{ \ \ \begin{array}{ll} \big| \mathrm H \big(\cdot,u,\mathrm D u, \ldots, \mathrm D^ku \big) \big|= \Lambda, & \text{ a.e.\ in }\Omega, \\ u = \varphi, & \text{ on }\partial \Omega,\\ \mathrm D u = \mathrm D \varphi, & \text{ on }\partial\Omega, \vdots & \vdots \\ \mathrm D^{k-1}u = \mathrm D^{k-1}\varphi, & \text{ on }\partial\Omega. \end{array} \right. \] Under appropriate assumptions for $\mathrm H$, we show existence of infinitely-many solutions $(u,\Lambda) \in \mathrm W^{k,\infty}_\varphi(\Omega;\mathbb R^N) \times [\Lambda_*,\infty)$ for $\Lambda_*\geq0$, by utilising the Baire Category method for implicit PDEs. In the case of $k=1$ and $n=N$, these assumptions do not require quasiconvexity.