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.

Table of Contents

• 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

• 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