Gradient estimates for solutions to divergence form elliptic equations with discontinuous coefficients.

*(English)*Zbl 0958.35060The purpose of this paper is to establish global \(W^{1,\infty}\) and piecewise \(C^{1,\alpha}\) estimates for solutions to divergence form elliptic equations with piecewise Hölder continuous coefficients. This work is motivated by the study of composite media with closely spaced interfacial boundaries. A standard model in this direction is the equation \(\partial_i(a(x)\partial_iu)=0\) in \(D\), with Dirichlet boundary condition on \(\partial D\), where \(D\) is a bounded open set in \({\mathbb{R}}^N\). The function \(u\) represents the out of plane elastic displacement, while \(a(x)=a_0\) for \(x\) inside the subdomains representing the fibers, and \(a(x)=1\) elsewhere in \(D\).

The novelty of the estimates given in the paper is that, even though they depend on the shape and on the size of the surfaces of discontinuity of the coefficients, they are independent of the distance between these surfaces. For instance, in the case of \(C^{1,\alpha'}\) interior estimates, the authors show that the restriction of the solution to each subdomain may be extended as a \(C^{1,\alpha'}\) function, with a norm that is independent of the distances between the subdomain interfaces. The proofs are based on elliptic regularity estimates (De Giorgi-Nash, Cordes-Nirenberg) and on perturbation arguments.

The reviewer considers that the present paper is a reference work, with deep implications in the next studies related to solutions of divergence form elliptic equations.

The novelty of the estimates given in the paper is that, even though they depend on the shape and on the size of the surfaces of discontinuity of the coefficients, they are independent of the distance between these surfaces. For instance, in the case of \(C^{1,\alpha'}\) interior estimates, the authors show that the restriction of the solution to each subdomain may be extended as a \(C^{1,\alpha'}\) function, with a norm that is independent of the distances between the subdomain interfaces. The proofs are based on elliptic regularity estimates (De Giorgi-Nash, Cordes-Nirenberg) and on perturbation arguments.

The reviewer considers that the present paper is a reference work, with deep implications in the next studies related to solutions of divergence form elliptic equations.

Reviewer: Vicenţiu D.Rădulescu (Craiova)