Localization coefficients of functions with applications in partial differential equations
Abstract
Abstract We identify shortcomings in two popular measures of localization of functions: the $$L^p\text {-}L^q$$ L p - L q participation ratio and the mass concentration comparison. We then introduce a novel localization measure for functions on bounded subsets of $$\varvec{R}^d$$ R d , $$d=1,2,3,\dots $$ d = 1 , 2 , 3 , ⋯ , based on a Wasserstein metric. For efficient computation, we prove the equality of this measure with a suitable Sobolev norm in dimension one. We demonstrate our approach by numerical experiments in one and two dimensions. Finally, we discuss and mitigate challenges arising from boundary effects.
Fields of Study & Subjects
Geometric Analysis and Curvature Flows
Nonlinear Partial Differential Equations
Numerical methods in inverse problems
Publication details
- Journal / source
- Partial Differential Equations and Applications
- Publisher
- Springer Science and Business Media LLC
- Volume
- 7
- Issue
- 5
- Publication date
- 12 September 2026
- Language
- EN