Nonlinear Partial Differential Equations

(Principal investigator: Mikko Parviainen)

The fundamental works of Doob, Hunt, Itô, Kakutani, Kolmogorov, Lévy and many others have shown a profound and powerful connection between the classical linear partial differential equations and probability. The interplay between linear PDEs and stochastics arises from the fact that harmonic functions and martingales share a common origin in mean value properties. The linear theory has several real-world applications such as mathematical option pricing and portfolio management, but the connection between PDEs and probability has also been useful in several breakthroughs in pure mathematics, for example in the celebrated regularity proof of Krylov and Safonov.

A similar connection has recently been established in the nonlinear theory: a class of two-player zero-sum stochastic games sometimes referred to as tug-of-games lead in the simplest case to the normalized p-Laplace and normalized p-parabolic type equations. Since the equations are in nondivergence form, the solutions are understood in the viscosity sense. We develop new techniques based on the interplay of PDEs and stochastic game theory. Our problems are related to regularity, existence and uniqueness both for the value functions of the games as well as the solutions of the PDEs. This is partly in collaboration with the stochastics group, University of Jyväskylä.

(Principal investigator: Tuomas Orponen)

Boundary value problems for elliptic PDEs, such as the Laplace equation, are fairly well understood. For example, it has been known since the 70s that the Laplace equation in a Lipschitz domain with L2-boundary values has a unique solution. Similar problems remain widely open for sub-elliptic PDEs. A basic example of a sub-elliptic partial differential operator is the Kohn-Laplacian. Even for this model sub-elliptic operator, boundary-value problems are poorly understood, at least compared with elliptic, or even parabolic, counterparts. The intrinsic geometry of the Kohn-Laplacian is not Euclidean, and studying the Kohn-Laplacian invites the researcher to the twisty realms of the Heisenberg group.

(Principal investigator: Vesa Julin)

By shape optimization, one usually refers to problems in Calculus of Variations where the minimization problem is among sets, not among functions. A classical example is the isoperimetric problem, which states that ball has the smallest surface area among all sets with same volume. This problem was known already by the ancient Greeks but was solved by an Italian mathematician E. De Giorgi only in 1950’s.
Today we are trying to understand the stability of the well-known isoperimetric and functional inequalities. We are interested to know what happens to the minimizer, say, of the isoperimetric problem when there are additional external forces affecting the set. An example of a much-studied model is the Ohta-Kawasaki functional, which is roughly the isoperimetric problem with an additional Columbic-interaction among particles. This functional is a fundamental model in material and in nuclear physics.

(Principal investigators: Petri Juutinen and Mikko Parviainen)

In order to understand the dynamics of nature, we need to consider time evolution, which often leads to parabolic partial differential equations. Perhaps the most well-known linear parabolic partial differential equation is the linear heat equation. However, many applications lead to the nonlinear parabolic partial differential equations. Further, nonlinear models intoduce new interesting phenomena from intrinsic behaviour to extinction in finite time. We study normalised p-parabolic equations arising from stochastic game theory, degenerate and singular p-parabolic type equations, limiting cases (infinity parabolic equation, mean curvature flow equation), extensions to systems as well as the porous medium equation. The techniques needed to tackle the problems are based on the viscosity solutions, distributional weak solutions, and stochastic game theory.

(Principal investigator: Tero Kilpeläinen)

Solutions to second order quasilinear elliptic equations have many properties in common with harmonic functions even if the principle of superposition is lost. A basic example is the p-Laplace operator which adopts the position of the Laplace operator in the nonlinear theory.
The research theme is to use ideas and methods from nonlinear potential theory (presented for example in the book by Heinonen, Kilpeläinen, and Martio; see below) to treat problems in PDEs. Nowadays, the main PDE problems considered deal with p-Laplace type equations involving measures. Equations are interpreted in the sense of distributions, but especially in the case of singular measures, one must carefully define what is meant by a solution. There are three types of natural questions under research:

• In which function class the equation has a unique solution for a given measure?
• If the measure is good, how large the unique solvability class may be?
• Find reasonable estimates for solutions and their regularity?

(Principal investigators: Petri Juutinen)

Contrary to the traditional variational problems where one typically seeks to minimize a certain weighted average of a quantity, the L^∞-problems relate to applications in which it is important to control extreme pointwise values. Concrete examples can be found for example in the optimal shape design with non-elastic materials, landslide modeling, and brain and surface warping.
A great deal of research has been devoted to understanding the special case of the minimal Lipschitz extension problem and the associated infinity Laplace equation . This problem can be approached by taking the limit as of the problems of finding a minimal p-extension, and thus there is a natural connection with the theory of p-Laplace type equations. However, the -variational problems differ from the classical ones in many respect, and thus genuinely new methods and tools are needed in studying their properties.