TY - JOUR

T1 - The insulated conductivity problem, effective gradient estimates and the maximum principle

AU - Weinkove, Ben

N1 - Funding Information:
Research supported in part by NSF Grant DMS-2005311. Part of this work was carried out while the author was visiting the Department of Mathematical Sciences at the University of Memphis whom he thanks for their kind support and hospitality.
Publisher Copyright:
© 2021, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.

PY - 2023/2

Y1 - 2023/2

N2 - We consider the insulated conductivity problem with two unit balls as insulating inclusions, a distance of order ε apart. The solution u represents the electric potential. In dimensions n≥ 3 it is an open problem to find the optimal bound on the gradient of u, the electric field, in the narrow region between the insulating bodies. Li-Yang recently proved a bound of order ε-(1-γ)/2 for some γ> 0. In this paper we use a direct maximum principle argument to sharpen the Li-Yang estimate for n≥ 4. Our method gives effective lower bounds on γ, which in particular approach 1 as n tends to infinity.

AB - We consider the insulated conductivity problem with two unit balls as insulating inclusions, a distance of order ε apart. The solution u represents the electric potential. In dimensions n≥ 3 it is an open problem to find the optimal bound on the gradient of u, the electric field, in the narrow region between the insulating bodies. Li-Yang recently proved a bound of order ε-(1-γ)/2 for some γ> 0. In this paper we use a direct maximum principle argument to sharpen the Li-Yang estimate for n≥ 4. Our method gives effective lower bounds on γ, which in particular approach 1 as n tends to infinity.

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U2 - 10.1007/s00208-021-02314-3

DO - 10.1007/s00208-021-02314-3

M3 - Article

AN - SCOPUS:85123479974

SN - 0025-5831

VL - 385

SP - 1

EP - 16

JO - Mathematische Annalen

JF - Mathematische Annalen

IS - 1-2

ER -