2025/04/04 by Katzourakis, Nikos · 1 citation
#35A15 #35J60 #Analysis of PDEs (math.AP) #FOS: Mathematics #Primary 35J47 #Secondary 35D30
paper · doi:10.48550/arxiv.2504.03972
Let Ω\Subset \mathbb Rn and a continuous function \mathrm H be given, where n,k,N ∈ \mathbb N. For p∈ [1,∞], we consider the functional \mathrm Ep(u) := ‖ \mathrm H (⋅,u,\mathrm D u, …, \mathrm Dku ) ‖\mathrm Lp(Ω), u∈ \mathrm Wk,p(Ω;\mathbb RN). We are interested in the L^∞ variational problem \mathrm C∞,p(u_∞) = inf \\mathrm C∞,p(u) : u∈ \mathrm Wk,∞φ(Ω;\mathbb RN), \mathrm E1(u)≠ 0 \, where φ∈ \mathrm Wk,∞(Ω;\mathbb RN), p is fixed, and \mathrm C∞,p(u) := (\mathrm E_∞(u))/(\mathrm Ep(u)) . The variational problem is ill-posed. \mathrm C∞,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 \ | \mathrm H (⋅,u,\mathrm D u, …, \mathrm Dku ) |= Λ, · amp; a.e. in Ω,
u = φ, · amp; on ∂ Ω,
\mathrm D u = \mathrm D φ, · amp; on ∂Ω,
⋮ · amp; ⋮
\mathrm Dk-1u = \mathrm Dk-1φ, · amp; on ∂Ω. . Under appropriate assumptions for \mathrm H, we show existence of infinitely-many solutions (u,Λ) ∈ \mathrm Wk,∞φ(Ω;\mathbb RN) × [Λ_*,∞) for Λ_*≥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.