This book is concerned with certain classes of nonlinear problems for elliptic systems of partial differential equations: boundary-value problems with free boundaries. The first part has to do with the general theory of boundary-value problems for analytic functions and its applications to hydrodynamics. The second presents the theory of quasiconformal mappings, along with the theory of boundary-value problems for elliptic systems of equations and applications of it to problems in the mechanics of continuous media with free boundaries: problems in subsonic gas dynamics, filtration theory, and problems in elastico-plasticity.
Part I Boundary-value problems in the theory of analytic functions and their application to hydrodynamics:; I. The boundary-value problems of Riemann and Hilbert; II. Singular operators in spaces of summable functions. Application to boundary-value problems and to the study of boundary properties of analytic functions; III. The mixed boundary-value problem with free boundary; IV. Flows of an incompressible fluid with free boundaries; Part II Generalized solutions of quasilinear systems of equations in the mechanics of continuous media:; V. Quasiconformal mappings and generalized solutions of elliptic systems of equations on the plane; VI. Boundary-value problems; VII. Boundary-value problems in hydrodynamics and subsonic gas dynamics; VIII. Problems in filtration theory for a fluid with free boundaries; IX. Some planar problems with an unknown boundary in elasticity theory.