Optimal boundary gradient estimates for Lamé systems with partially infinite coefficients

Abstract

In this paper, we derive the pointwise upper bounds and lower bounds on the gradients of solutions to the Lamé systems with partially infinite coefficients as the surface of discontinuity of the coefficients of the system is located very close to the boundary. When the distance tends to zero, the optimal blow-up rates of the gradients are established for inclusions with arbitrary shapes and in all dimensions.

Singular analysis of the stress concentration in the narrow regions between the inclusions and the matrix boundary

“…The mathematical approaches in [26] are based on the layer potential techniques and the variational principle, which are different from the iterate scheme above. Besides the aforementioned interior estimates, there is another direction of research to establish the boundary estimates [8,35]. For one strictly convex inclusion close to touching the matrix boundary, Bao, Ju and Li [8] obtained the optimal upper and lower bounds on the gradient.…”

“…Besides the aforementioned interior estimates, there is another direction of research to establish the boundary estimates [8,35]. For one strictly convex inclusion close to touching the matrix boundary, Bao, Ju and Li [8] obtained the optimal upper and lower bounds on the gradient. In particular, the lower bound on the gradient in [8] was constructed by finding a blow-up factor which is a linear functional in relation to the boundary data.…”

“…For one strictly convex inclusion close to touching the matrix boundary, Bao, Ju and Li [8] obtained the optimal upper and lower bounds on the gradient. In particular, the lower bound on the gradient in [8] was constructed by finding a blow-up factor which is a linear functional in relation to the boundary data. Subsequently, Li and Zhao [35] extended to the general m-convex inclusions and found that some special boundary data with k-order growth will strengthen the singularities of the stress.…”

Gradient estimates for the perfect conductivity problem in anisotropic media

“…When k degenerates (k → 0 or k → +∞) the scenario is very different: the gradient of the solution may be unbounded as δ → 0 and the blow-up rate depends on the dimension. Indeed, it has been proved that the optimal blow-up rate of |∇u| is δ −1/2 for N = 2, it is (δ| log δ|) −1 for N = 3 and δ −1 for N ≥ 4, see [1,2,3,6,7,8,9,10,11,26,27,28,30,31,35,32,42,43] and references therein.…”

Stress concentration for closely located inclusions in nonlinear perfect conductivity problems

“…(see for instance [6]). In the case of smooth inclusions, it has been proved that the optimal blow-up rate of |∇u| is δ −1/2 for N = 2, it is (δ| log δ|) −1 for N = 3 and δ −1 for N ≥ 4, see [1,2,3,5,6,7,8,9,10,21,22,23,24,28,29] and references therein. In addition to its mathematical interest, the characterization of the gradient blow-up is relevant for the applications in composite materials.…”