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- Title
The insulated conductivity problem, effective gradient estimates and the maximum principle.
- Authors
Weinkove, Ben
- Abstract
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.
- Subjects
ELECTRIC potential; UNIT ball (Mathematics); ELECTRIC fields; IONIC conductivity
- Publication
Mathematische Annalen, 2023, Vol 385, Issue 1/2, p1
- ISSN
0025-5831
- Publication type
Article
- DOI
10.1007/s00208-021-02314-3