This is a preliminary version of the first part of a book project that will consist of four parts. We are making it available in electronic form now, because there is a demand for some of the technical tools it provides, in particular a detailed presentation of analytic elliptic regularity estimates in the neighborhood of smooth boundary points. In our proof of analytic a priori estimates, besides the classical Morrey-Nirenberg techniques of nested open sets and difference quotients, a new ingredient is a Cauchy-type estimate for coordinate transformations based on the Faà di Bruno formula for derivatives of composite functions.
This first part can also serve as a general introduction into the subject of regularity for linear elliptic systems with smooth coefficients on smooth domains. We treat regularity in Sobolev spaces for a general class of second order elliptic systems and corresponding boundary operators that cover, in particular, many elliptic problems in variational form. Starting from the regularity of the variational solution, we follow the improvement of the regularity of the solution as the regularity of the data is raised, first for low regularity, and then going to ever higher regularity and finally to analytic regularity. Supported by the discussion of many examples, some of them new, such as the variational formulation of the electromagnetic impedance problem, we hope to provide new insight into this classical subject.
We hope to be able to finish the whole project soon and to publish all four parts, but in the meantime this first part can be used as a starting point for proofs of elliptic regularity estimates in more complicated situations.
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