Back to Search
Start Over
Optimal gradient estimates for the perfect conductivity problem with C1,α inclusions
- Source :
- Annales de l'Institut Henri Poincaré C, Analyse non linéaire. 38:953-979
- Publication Year :
- 2021
- Publisher :
- European Mathematical Society - EMS - Publishing House GmbH, 2021.
-
Abstract
- In high-contrast composite materials, the electric field concentration is a common phenomenon when two inclusions are close to touch. It is important from an engineering point of view to study the dependence of the electric field on the distance between two adjacent inclusions. In this paper, we derive upper and lower bounds of the gradient of solutions to the conductivity problem where two perfectly conducting inclusions are located very close to each other. To be specific, we extend the known results of Bao-Li-Yin (ARMA 2009) in two folds: First, we weaken the smoothness of the inclusions from C 2 , α to C 1 , α . To obtain a pointwise upper bound of the gradient, we follow an iteration technique which is first used to deal with elliptic systems in a narrow domain by Li-Li-Bao-Yin (QAM 2014). However, when the inclusions are of C 1 , α , we can not use W 2 , p estimates for elliptic equations any more. In order to overcome this new difficulty, we take advantage of De Giorgi-Nash estimates and Campanato's approach to apply an adapted version of the iteration technique with respect to the energy. A lower bound in the shortest line between two inclusions is also obtained to show the optimality of the blow-up rate. Second, when two inclusions are only convex but not strictly convex, we prove that blow-up does not occur any more. The establishment of the relationship between the blow-up rate of the gradient and the order of the convexity of the inclusions reveals the mechanism of such concentration phenomenon.
- Subjects :
- Pointwise
Smoothness (probability theory)
Applied Mathematics
010102 general mathematics
Mathematical analysis
Regular polygon
01 natural sciences
Upper and lower bounds
Domain (mathematical analysis)
Convexity
010101 applied mathematics
Line (geometry)
0101 mathematics
Convex function
Mathematical Physics
Analysis
Mathematics
Subjects
Details
- ISSN :
- 18731430 and 02941449
- Volume :
- 38
- Database :
- OpenAIRE
- Journal :
- Annales de l'Institut Henri Poincaré C, Analyse non linéaire
- Accession number :
- edsair.doi...........d537d0f63ce0ee173fee491a894a1f9c