The Variable Coefficient Hele-Shaw Problem, Integrability and Quadrature Identities

The theory of quadrature domains for harmonic functions and the Hele-Shaw problem of the fluid dynamics are related subjects of the complex variables and mathematical physics. We present results generalizing the above subjects for elliptic PDEs with variable coefficients, emerging in a class of the free-boundary problems for viscous flows in non-homogeneous media. Such flows possess an infinite number of conservation laws, whose special cases may be viewed as quadrature identities for solutions of variable-coefficient elliptic PDEs. If such PDEs are gauge equivalent to the Laplace equation (gauge-trivial case), a time-dependent conformal map technique, employed for description of the quadrature domains, leads to differential equations, known as “string” constraints in the theory of integrable systems. Although analogs of the string constraints have non-local forms for gauge-non-trivial equations, it is still possible to construct the quadrature domains explicitly, if the elliptic operator belongs to a class of the Calogero-Moser Hamiltonians.