On the Luxemburg Functional in the Concave Regime with an Application to Nonlocal Differential Equations

arXiv:2610.35878v1 Announce Type: new Abstract: For $p:[0,1]\to(0,1)$ a continuous function, we consider the Luxemburg functional \[ \Lambda_{p(\cdot)}(u):=\inf\left\{\delta>0:\int_0^1\left|\frac{u(t)}{\delta}\right|^{p(t)}\,dt\leq 1\right\}. \] We develop a suitable functional analytic framework for $\Lambda_{p(\cdot)}$ 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 \[ -A\left(\int_0^1 b(1-s)(u(s))^{p(s)}\,ds\right)u''(t) =\lambda f(t,u(t)),\qquad t\in(0,1). \] 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.

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