On the Luxemburg Functional in the Concave Regime With an Application to Nonlocal Differential Equations
Year: 2026
Type: preprint
Abstract: For $p:[0,1]\to(0,1)$ a continuous function, we consider the Luxemburg functional \[ Λ_{p(\cdot)}(u):=\inf\left\{δ>0:\int_0^1\left|\frac{u(t)}δ\right|^{p(t)}\,dt\leq 1\right\}. \] We develop a suitable functional analytic framework for $Λ_{p(\cdot)}$ so that it can be incorporated into a positive order cone, which can then be used in the analysis of nonlocal differential equations having the general form \[ -A\left(\int_0^1 b(1-s)(u(s))^{p(s)}\,ds\right)u''(t) =λf(t,u(t)),\qquad t\in(0,1). \] By means of a specific numerical example, we demonstrate that, as opposed to a methodology relying only on the supremum norm, our methodology via the Luxemburg functional yields an appreciably sharper localisation of any solutions of the nonlocal equation.
Source: arXiv (Cornell University)
Authors: Christopher S Goodrich, Gabriel Nakhl
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7202395342