On the minimisation of the Peak-to-average ratio
Nikos Katzourakis
TL;DR
The paper addresses the ill-posed problem of minimising the Crest factor $C_{\infty,p}(u)=E_\infty(u)/E_p(u)$ for higher-order supremal functionals, with $E_p(u)=\|H(\cdot,u,Du,\ldots,D^k u)\|_{L^p(\Omega)}$ and $E_\infty(u)=\|H(\mathrm D^{[k]}u)\|_{L^\infty(\Omega)}$. It shows that minimisers are exactly strong a.e. solutions to a fully nonlinear eigenvalue Dirichlet problem $|H(\mathrm D^{[k]}u)|=\Lambda$ a.e. in $\Omega$ with boundary data up to order $k-1$, for some $\Lambda>0$, and proves the existence of infinitely many pairs $(u,\Lambda)$ via the Dacorogna–Marcellini Baire Category method. The results hold under structural assumptions such as Morrey quasiconvexity (or, in the special case $k=1$, $n=N$, a solvable singular-value auxiliary problem without requiring quasiconvexity), providing a direct link between peak-to-average control and nonlinear PDE eigenvalue problems. Importantly, the approach avoids relaxation and Young-measure techniques, offering a constructive framework for minimisers in higher-order $L^\infty$ variational problems and highlighting potential applications in signal processing and reactor modelling where uniform pointwise energy is desirable.
Abstract
Let $Ω\Subset \mathbb R^n$ and a continuous function $\mathrm H$ be given, where $n,k,N \in \mathbb N$. For $p\in [1,\infty]$, we consider the functional \[ \mathrm E_p(u) := \big\| \mathrm H \big(\cdot,u,\mathrm D u, \ldots, \mathrm D^ku \big) \big\|_{\mathrm L^p(Ω)},\ \ \ u\in \mathrm W^{k,p}(Ω;\mathbb R^N). \] We are interested in the $L^\infty$ variational problem \[ \mathrm C_{\infty,p}(u_\infty)\, =\, \inf \Big\{\mathrm C_{\infty,p}(u) \ : \ u\in \mathrm W^{k,\infty}_\varphi(Ω;\mathbb R^N), \ \mathrm E_1(u)\neq 0 \Big\}, \] where $\varphi\in \mathrm W^{k,\infty}(Ω;\mathbb R^N)$, $p$ 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 \[ \left\{ \ \ \begin{array}{ll} \big| \mathrm H \big(\cdot,u,\mathrm D u, \ldots, \mathrm D^ku \big) \big|= Λ, & \text{ a.e.\ in }Ω, \\ u = \varphi, & \text{ on }\partial Ω,\\ \mathrm D u = \mathrm D \varphi, & \text{ on }\partialΩ, \vdots & \vdots \\ \mathrm D^{k-1}u = \mathrm D^{k-1}\varphi, & \text{ on }\partialΩ. \end{array} \right. \] Under appropriate assumptions for $\mathrm H$, we show existence of infinitely-many solutions $(u,Λ) \in \mathrm W^{k,\infty}_\varphi(Ω;\mathbb R^N) \times [Λ_*,\infty)$ for $Λ_*\geq0$, by utilising the Baire Category method for implicit PDEs. In the case of $k=1$ and $n=N$, these assumptions do not require quasiconvexity.