TY - JOUR

T1 - Minimizing Optimal Transport for Functions with Fixed-Size Nodal Sets

AU - Du, Qiang

AU - Sagiv, Amir

N1 - Publisher Copyright:
© 2023, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.

PY - 2023/10

Y1 - 2023/10

N2 - Consider the class of zero-mean functions with fixed L∞ and L1 norms and exactly N∈ N nodal points. Which functions f minimize Wp(f+, f-) , the Wasserstein distance between the measures whose densities are the positive and negative parts? We provide a complete solution to this minimization problem on the line and the circle, which provides sharp constants for previously proven “uncertainty principle”-type inequalities, i.e., lower bounds on N· Wp(f+, f-) . We further show that, while such inequalities hold in many metric measure spaces, they are no longer sharp when the non-branching assumption is violated; indeed, for metric star-graphs, the optimal lower bound on Wp(f+, f-) is not inversely proportional to the size of the nodal set, N. Based on similar reductions, we make connections between the analogous problem of minimizing Wp(f+, f-) for f defined on Ω ⊂ Rd with an equivalent optimal domain partition problem.

AB - Consider the class of zero-mean functions with fixed L∞ and L1 norms and exactly N∈ N nodal points. Which functions f minimize Wp(f+, f-) , the Wasserstein distance between the measures whose densities are the positive and negative parts? We provide a complete solution to this minimization problem on the line and the circle, which provides sharp constants for previously proven “uncertainty principle”-type inequalities, i.e., lower bounds on N· Wp(f+, f-) . We further show that, while such inequalities hold in many metric measure spaces, they are no longer sharp when the non-branching assumption is violated; indeed, for metric star-graphs, the optimal lower bound on Wp(f+, f-) is not inversely proportional to the size of the nodal set, N. Based on similar reductions, we make connections between the analogous problem of minimizing Wp(f+, f-) for f defined on Ω ⊂ Rd with an equivalent optimal domain partition problem.

KW - Metric graph

KW - Nodal set

KW - Optimal partition

KW - Uncertainty principle

KW - Wasserstein

UR - http://www.scopus.com/inward/record.url?scp=85168310720&partnerID=8YFLogxK

U2 - 10.1007/s00332-023-09952-8

DO - 10.1007/s00332-023-09952-8

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AN - SCOPUS:85168310720

SN - 0938-8974

VL - 33

JO - Journal of Nonlinear Science

JF - Journal of Nonlinear Science

IS - 5

M1 - 95

ER -