The primary objective of this book is to give a comprehensive
exposition of results surrounding the work of the authors concerning
boundary regularity of weak solutions of second-order elliptic
quasilinear equations in divergence form. The structure of these
equations allows coefficients in certain $L^{p}$ spaces, and thus it
is known from classical results that weak solutions are locally
Hölder continuous in the interior. Here it is shown that weak
solutions are continuous at the boundary if and only if a Wiener-type
condition is satisfied. This condition reduces to the celebrated
Wiener criterion in the case of harmonic functions. The work that
accompanies this analysis includes the "fine" analysis of Sobolev
spaces and a development of the associated nonlinear potential
theory. The term "fine" refers to a topology of $\mathbf R^{n}$ which is
induced by the Wiener condition.
The book also contains a complete development of regularity of solutions of
variational inequalities, including the double obstacle problem, where the
obstacles are allowed to be discontinuous. The regularity of the solution is
given in terms involving the Wiener-type condition and the fine topology. The
case of differential operators with a differentiable structure and
$\mathcal C^{1,\alpha}$ obstacles is also developed. The book concludes with a chapter
devoted to the existence theory, thus providing the reader with a complete
treatment of the subject ranging from regularity of weak solutions to the
existence of weak solutions.
Readership
Graduate students and research mathematicians interested in the
theory of regularity of weak solutions of elliptic differential equations,
Sobolev space theory, and potential theory.