## Final Activity Report Summary - LG and IPP (Laplacian growth and Inverse Potential problems)

A number of interesting mathematical subjects grew from the problem of understanding interface dynamics and pattern formation. Being one of the most rapidly developing current themes in nonlinear science, the problem produces beautiful connections with different branches of fundamental mathematics, such as theory of quadrature domains, conformal mappings, Schwarz function, theory of dispersionless integrable hierarchies, potential theory, etc.

Recently, the integrable structures were found in a class of hydrodynamics' problems leading to a pattern formation in a regime far from equilibrium. Growth problems of this type were unified by the name Laplacian growth. Also known as the Hele-Shaw problem, it refers to dynamics of a moving front or interface between two distinct phases driven by a harmonic scalar field. This field is a potential for the growth velocity field. The Laplacian growth problem appears in different physical and mathematical contexts and has a number of important applications.

On an initial project stage we studied a class of solutions to Laplacian growth, being finite-dimensional reductions of the related dispersionless hierarchy constrained by 'string equation'. We considered polynomial, rational and logarithmic reductions and established their Hamiltonian structure.

In a course of further project development, we found new integrable generalisations of the Laplacian growth problem, the so-called variable-coefficient Hele-Shaw problem. Those were inhomogeneous flows through porous medium driven by multipole sources. They were no more described by Laplacian but rather by some partial differential equation of elliptic type. Therefore, we extended the class of exactly solvable Laplacian growth problems by adding a wide class of non-Laplacian growth problems and providing new important examples related to quantum integrable systems.

The impact of the research was widely interdisciplinary. New explicit solutions for variable-coefficient Hele-Shaw problem in the process of derivation offered the possibility to construct a new class of quadrature domains for non-harmonic functions, the so-called generalised quadrature domains. By definition, quadrature domains were domains described by quadrature relations and the latter were such integral relations that function under integral was evaluated by taking its values in a finite number of points inside a quadrature domain. The most known example of quadrature relation is the mean value theorem for a harmonic function, when the integral of a unit circle of a harmonic function is evaluated by taking its value in the centre. Quadrature domains arose here as conservation laws for the variable-coefficient Hele-Shaw problem and were described by generalised quadrature identities involving integral of non-harmonic functions.

On the other side, the class of possible partial differential equations of elliptic type for which exact solutions could be constructed were used in the classification of the so-called Huygensian potentials. Those were potentials that described waves possessing the Huygens property in Hadamard precise. The main example of Huygens property is the property of sound waves, emitted when we are speaking, to possess a strict first and rear fronts. We conjectured that partial differential equations with Huygensian potentials exhausted the corresponding list of exactly solvable variable-coefficient Hele-Shaw problems.

During the Banff conference on quadrature domains and Laplacian growth in Modern Physics, July 2007, it was proposed to distinguish the investigated class of problems to a special direction called 'elliptic growth'.

Recently, the integrable structures were found in a class of hydrodynamics' problems leading to a pattern formation in a regime far from equilibrium. Growth problems of this type were unified by the name Laplacian growth. Also known as the Hele-Shaw problem, it refers to dynamics of a moving front or interface between two distinct phases driven by a harmonic scalar field. This field is a potential for the growth velocity field. The Laplacian growth problem appears in different physical and mathematical contexts and has a number of important applications.

On an initial project stage we studied a class of solutions to Laplacian growth, being finite-dimensional reductions of the related dispersionless hierarchy constrained by 'string equation'. We considered polynomial, rational and logarithmic reductions and established their Hamiltonian structure.

In a course of further project development, we found new integrable generalisations of the Laplacian growth problem, the so-called variable-coefficient Hele-Shaw problem. Those were inhomogeneous flows through porous medium driven by multipole sources. They were no more described by Laplacian but rather by some partial differential equation of elliptic type. Therefore, we extended the class of exactly solvable Laplacian growth problems by adding a wide class of non-Laplacian growth problems and providing new important examples related to quantum integrable systems.

The impact of the research was widely interdisciplinary. New explicit solutions for variable-coefficient Hele-Shaw problem in the process of derivation offered the possibility to construct a new class of quadrature domains for non-harmonic functions, the so-called generalised quadrature domains. By definition, quadrature domains were domains described by quadrature relations and the latter were such integral relations that function under integral was evaluated by taking its values in a finite number of points inside a quadrature domain. The most known example of quadrature relation is the mean value theorem for a harmonic function, when the integral of a unit circle of a harmonic function is evaluated by taking its value in the centre. Quadrature domains arose here as conservation laws for the variable-coefficient Hele-Shaw problem and were described by generalised quadrature identities involving integral of non-harmonic functions.

On the other side, the class of possible partial differential equations of elliptic type for which exact solutions could be constructed were used in the classification of the so-called Huygensian potentials. Those were potentials that described waves possessing the Huygens property in Hadamard precise. The main example of Huygens property is the property of sound waves, emitted when we are speaking, to possess a strict first and rear fronts. We conjectured that partial differential equations with Huygensian potentials exhausted the corresponding list of exactly solvable variable-coefficient Hele-Shaw problems.

During the Banff conference on quadrature domains and Laplacian growth in Modern Physics, July 2007, it was proposed to distinguish the investigated class of problems to a special direction called 'elliptic growth'.