This book investigates the behavior of weak or strong solutions of the
boundary value problems for the second order elliptic equations (linear and
quasilinear) in the neighborhood of the boundary singularities. The author's
main goal is to establish the precise exponent of the solution decrease rate
and under the best possible conditions.
Currently there exists a fully developed theory for linear elliptic equations
with partial derivatives so that it is now possible to advance toward
a nonlinear equations analysis. Considerable success in this direction has
been achieved particularly for the second order quasilinear elliptic equations,
due to the works of Schauder, Caccioppoli, Leray and others (see
[43, 129, 211, 216]). These authors have shaped a method that allows
to prove existence theorems given the appropriate a priori estimates. This
method does not require preliminary construction of the fundamental solution
and allows instead an application of some functional analysis theorems
rather than using an integral equation theory.