Young functions on varifolds. Part I. Functional analytic foundations

The series of papers is devoted to the study of convergence for pairs of surfaces and smooth functions thereon. We model such pairs with varifolds and multiple-valued functions to capture their limits. In the present paper, we study Young functions, a measure-theoretic approach to multiple-valued functions, and the graph measures associated with pairs of measures (in particular, varifolds) and Young functions. This setting allows us to model the convergence of pairs of surfaces and functions thereon via the weak convergence of their associated graph measures, and a compactness theorem follows immediately. As a prerequisite for the concepts of differentiability for Young functions in the upcoming papers, we introduce and investigate several test function spaces.

💡 Research Summary

The paper “Young functions on varifolds. Part I. Functional analytic foundations” develops a new analytical framework for studying the joint convergence of surfaces (modeled as varifolds) and functions defined on them, especially when the functions may take multiple values or diffuse in the limit. Classical Q‑valued function theory (Almgren, De Lellis–Spadaro) assumes a fixed integer multiplicity Q and therefore cannot accommodate the more intricate behavior that arises when a sequence of varifolds converges: the number of values of the limiting function may vary from point to point, and the values may spread out rather than remain discrete.

To overcome these limitations the author introduces Young functions. Given locally compact Hausdorff spaces X (the base) and Y (the target) and a Radon measure μ on X, a μ‑Young function f of type Y is a μ‑measurable map x↦f(x) taking values in the space P(Y) of probability Radon measures on Y. P(Y) is equipped with the initial topology generated by the linear functionals ν↦∫k dν for all continuous compactly supported k∈C_c(Y). Ordinary single‑valued measurable maps g:X→Y correspond to Young functions via f(x)=δ_{g(x)}; similarly, Q‑valued functions give rise to Young functions after a measurable selection. Crucially, Young functions allow diffuse measures and pointwise varying multiplicities.

Associated to a pair (μ,f) the author defines a graph measure Y(μ,f) on X×Y by the rule
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