Optimal gradient estimates of solutions to the insulated conductivity problem in dimension greater than two

  • Hongjie Dong

    Brown University, Providence, USA
  • YanYan Li

    Rutgers University, Piscataway, USA
  • Zhuolun Yang

    Brown University, Providence, USA
Optimal gradient estimates of solutions to the insulated conductivity problem in dimension greater than two cover

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Abstract

We study the insulated conductivity problem with inclusions embedded in a bounded domain in . The gradient of solutions may blow up as , the distance between inclusions, approaches to . It was known that the optimal blow-up rate in dimension is of order . It has recently been proved that in dimensions , an upper bound of the gradient is of order for some . On the other hand, optimal values of  have not been identified. In this paper, we prove that when the inclusions are balls, the optimal value of  is in dimensions .

Cite this article

Hongjie Dong, YanYan Li, Zhuolun Yang, Optimal gradient estimates of solutions to the insulated conductivity problem in dimension greater than two. J. Eur. Math. Soc. (2024), published online first

DOI 10.4171/JEMS/1432