The Dirichlet problem with \(L^ 2\)-boundary data for elliptic linear equations
It is considered the Dirichlet problem with \(L^ 2\)-boundary data for linear second order elliptic equations of the form \[ -D_ i(a^{ij}(x)D_ ju(x)+b^ i(x)u(x))+d^ i(x)D_ iu(x)+c(x)u(x)=f(x)\text{ in } Q, \] \[ u(x)=\phi (x)\text{ on } \partial Q \] where Q is a (sufficiently regular) bounded domain in \(R_ n\). Here \(\phi\) is assumed to be in \(L^ 2(\partial Q)\), therefore we cannot expect to find a solution in the space \(W^{1,2}(Q)\), since not every function in \(L^ 2(\partial Q)\) is the trace of a function of \(W^{1,2}(Q)\). This difficulty is overcome by introducing a suitable weighted Sobolev space \(\tilde W^{1,2}(Q)\) instead of \(W^{1,2}(Q)\), and by considering traces of the solution on surfaces parallel to \(\partial Q.\) It would be too long to list all the results of this work, which involves also reverse Hölder inequalities, harmonic measures, degenerate ellipticity, Carleson measures. The hypotheses on the boundary \(\partial Q\) and on the coefficients \(a^{ij}\) are often more general than in previous papers on the same subject. In conclusion, the book is almost self contained, well written, up-to- date and it will certainly interest the numerous specialists on elliptic equations.
scientific article; zbMATH DE number 51253