Gradient estimates for elliptic systems from composite materials with closely spaced stiff $C^{1,\gamma}$ inclusions

TL;DR

This paper derives pointwise gradient bounds for elliptic systems with Hölder continuous coefficients in narrow regions, aiding understanding of damage initiation in composite materials with closely spaced stiff inclusions.

Contribution

It establishes new gradient estimates under weaker boundary regularity conditions, extending previous results to $C^{1,eta}$ boundaries for damage analysis.

Findings

Gradient bounds indicate damage can initiate at the narrowest regions.

Estimates hold for systems with $C^{1,eta}$ boundaries, weaker than $C^{2,eta}$.

Results are applicable to composite material damage modeling.

Abstract

This paper is devoted to establishing the pointwise upper and lower bounds estimates of the gradient of the solutions to a class of general elliptic systems with H\"{o}lder continuous coefficients in a narrow region where the upper and lower boundaries is $C^{1,\gamma},$ $0<\gamma<1$, weaker than the previous $C^{2,\gamma}$ assumption. These estimates play a key role in the damage analysis of composite materials. From our results, the damage may initiate from the narrowest place.