This paper, somewhat delayed in the series “Fundamental Directions,” is devoted to general linear elliptic boundary problems on a smooth compact manifold with boundary. The paper is intended for a large circle of readers. We hope the paper will be useful to many mathematicians with diverse scientific interests. The main features of the theory we will discuss were formed in the 60’s, beginning with the proof of the equivalence of the ellipticity conditions and the Fredholm property of the corresponding operator in the simplest Sobolev spaces. This was done on the basis of achievements of many mathematicians during the preceding decades. This basis was very extensive, but the results were incomplete. The elaboration of the general theory was stimulated by investigations of the index problem (see e.g. (Palais 1965) and (Fedosov 1990)) and went on under a strong influence of the microlocal analysis, beginning with the appearance of the calculus of pseudodifferential operators. During the last decades, new variants of the general theory appeared, and the old variants were enriched by new results.

Elliptic Boundary Problems

The Lp boundary value problems on Lipschitz domains

We consider second order elliptic divergence form systems with complex measurable coefficients A that are independent of the transversal coordinate, and prove that the set of A for which the boundary value problem with L2 Dirichlet or Neumann data is well posed, is an open set. Furthermore we prove that these boundary value problems are well posed when A is either Hermitean, block or constant. Our methods apply to more general systems of partial differential equations and as an example we prove perturbation results for boundary value problems for differential forms.

/pdf/solvability-of-elliptic-systems-with-square-integrable-4hmui4utms.pdf

Solvability of elliptic systems with square integrable boundary data

https://projecteuclid.org/download/pdf_1/euclid.acta/1485891731

The biharmonic Neumann problem in Lipschitz domains

In this paper we use the method of layer potentials to study L 2 boundary value problems in a bounded Lipschitz domain Ω for a family of second-order elliptic systems with rapidly oscillating periodic coefficients. Defining L e = -div(A(e -1 X)∇), under the assumption that A(X) is elliptic, symmetric, periodic, and Holder-continuous, we establish the solvability of the L 2 Dirichlet, regularity, and Neumann problems for L e (u e ) = 0 in Ω with optimal estimates uniform in e > 0.

Wenjie Gao

Papers

Layer potentials and boundary value problems for elliptic systems in Lipschitz domains