Let \(\Omega \Subset \mathbb R^n\) and a continuous function \(\textrm{H}: \Omega \times \mathbb R^N \times \mathbb R^{N n} \times \cdots \times \mathbb R^{N n^{k}}_s \longrightarrow \mathbb R\) be given, where \(n,k,N \in \mathbb N\) . For \(p\in [1,\infty ]\) , we consider the functional \(\begin{aligned} \ \ \ \ \ \mathrm E_p(u) := \big \Vert \mathrm H \big (\cdot ,u,\mathrm D u, \ldots , \mathrm D^ku \big ) \big \Vert _{\mathrm L^p(\Omega )},\ \ \ u\in \mathrm W^{k,p}(\Omega ;\mathbb R^N). \end{aligned}\) In this note we are interested in the \(\textrm{L}^\infty \) variational problem \(\begin{aligned} \ \ \ \ \ \mathrm C_{\infty ,p}(u_\infty )\, =\, \inf \Big \{\mathrm C_{\infty ,p}(u) \ : \ u\in \mathrm W^{k,\infty }_\varphi (\Omega ;\mathbb R^N), \ \mathrm E_1(u)\ne 0 \Big \},\qquad \qquad \qquad (*) \end{aligned}\) where \(\varphi \in \mathrm W^{k,\infty }(\Omega ;\mathbb {R}^N)\) is given, p is fixed, and \(\begin{aligned} \mathrm C_{\infty ,p}(u)\, := \, \frac{\mathrm E_\infty (u)}{\mathrm E_p(u)} . \end{aligned}\) 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 ( \(*\) ) by characterising the set of minimisers as the set of strong solutions to the eigenvalue Dirichlet problem for the fully nonlinear PDE \(\begin{aligned}\left\{ \ \ \begin{array}{ll} \big | \mathrm H \big (\cdot ,u,\mathrm D u, \ldots , \mathrm D^ku \big ) \big |= \Lambda , & \text { a.e. in }\Omega , \\ u = \varphi , & \text { on }\partial \Omega ,\\ \mathrm D u = \mathrm D \varphi , & \text { on }\partial \Omega ,\\ \ \ \ \ \vdots \ \ \ & \ \ \ \ \ \vdots \ \ \ \ \\ \mathrm D^{k-1}u = \mathrm D^{k-1}\varphi , & \text { on }\partial \Omega . \end{array} \right. \qquad \qquad \qquad (**) \end{aligned}\) Under appropriate assumptions for \(\textrm{H}\) , we show existence of infinitely-many solutions \((u,\Lambda ) \in \mathrm W^{k,\infty }_\varphi (\Omega ;\mathbb R^N) \times [\Lambda _*,\infty )\) to ( \(**\) ) for \(\Lambda _*\ge 0\) , by utilising the Baire Category method for implicit PDEs. In the case of \(k=1\) and \(n=N\) , these assumptions do not require quasiconvexity.
