Regularity Of Solutions To Elliptic Equations (Systems) From Composite Material

In the paper, we consider three elliptic problems from composite material: we firstly consider a elliptic system in a cube in R" which has lower order terms, the coefficientsare smooth in the closures of subdomains but possibly discontinuous across their boundaries, in Chapter 2 we research the regularity of its solution; In Chapter 3, we mainly study the conductivity problem where two circular conductivity inclusions are very close but not touching, we make use of the theory of the layer popential, and establish the normal derivative estimates for the solution ; Finally, we further consider the perfect conductivity problem in all dimensions, we establish lower bounds of the gradient estimates for the solution, and know whether the gradient can be arbitrarily large as the inclusions get closer to each other.The paper is made up of five parts as follows:In Chapter 1, we introduce the development of the composite material, the backgroundof selecting this question and the significance in both theory and practice.In Chapter 2, we consider a composite material in R" whena cubeΩ= {x : |x_i| <1} contains L disjoint subdomainsΩ_m = {x∈Ω: c_m-1 < x_n < c_m}, 1≤m≤L, c_m are increasing constants lying between -1 and 1. We suppose vector-valued functions u(x) = (u¹(x),…,u^N(x)) is the solution to elliptic systems as follows:where A_ij^αβ,B_ij^β, C_ij, H_i,G_i^α∈C∞(?)(m = 1,2,…, L). We prove its weak solution u∈C^∞(?∩Ω), and establish the upper estimate for its solution.In Chapter 3, we mainly study the conductivity problem where two circular conductivityinclusions B₁ and B₂ are very close but not touching:where H(x) is a harmonic function in R². For i=1,2, we suppose that the conductivity k_i of the inclusions B_i is a constant different from 1. By the theory of the layer potential, we decompose the solution of the problem, then establish the normal derivative estimate for the solution. In Chapter 4, we further study the perfect conductivity problem which is a particularcase that the conductivity of the inclusions is∞:where H(x) satisfy (?) = 0 in Rⁿ. We give the accurate blow-up rates of the gradient for the solution in all dimensions.