# Nonlinear Partial Differential Equations

Partial differential equations have a great variety of applications to mechanics, electrostatics, quantum mechanics and many other fields of physics as well as to finance. In the linear theory, solutions obey the principle of superposition and they often have representation formulas. However, it is sometimes said that the great discovery of the 19th century was that the equations of nature are linear whereas the great discovery of the 20th century was that they are not. Nonlinear PDEs appear for example in stochastic game theory, non-Newtonian fluids, glaceology, rheology, nonlinear elasticity, flow through a porous medium, and image processing. Since superposition is not available, methods needed to study nonlinear equations are quite different from those of the linear theory.

Our research is based on active collaboration both nationally and internationally. The group invites several collaborators to visit Finland each year, and has interesting research topics for starting researchers.

## People

Professor Petri Juutinen

Professor Tero Kilpeläinen

Academy research fellow Mikko Parviainen

Postdoctoral researcher Amal Attouchi

Postdoctoral researcher Vesa Julin

Postdoctoral researcher Hannes Luiro

Postdoctoral researcher Eero Ruosteenoja

Some former members: Finland Distinguished Professor Nicola Fusco, professor Xiao Zhong

group also has several PhD students

## Research

### Parabolic PDEs

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 investigators: Petri Juutinen and Mikko Parviainen

### Nonlinear partial differential equations and their counterpart in stochastic game theory

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: Mikko Parviainen

### Shape Optimization

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 investigator: Vesa Julin

### Quasilinear elliptic partial differential equations

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 investigator: Tero Kilpeläinen

### Calculus of Variations in

Contrary to the traditional variational problems where one typically seeks to minimize a certain weighted average of a quantity, the -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.

Principal investigator: Petri Juutinen

## Publications

Publications of the group can be found on the arXiv preprint server and the TUTKA database. The most up-to-date information is available on the members' personal web pages.

### Selected publications

- H. Luiro, M. Parviainen, and E. Saksman. Harnack's inequality for p-harmonic functions via stochastic games. Comm. Partial Differential Equations. 38(12):1985-2003, 2013.
- S. Armstrong, M. Crandall, V. Julin, and C. Smart. Convexity criteria and uniqueness of absolutely minimizing functions. Arch. Ration. Mech. Anal. 200(2): 405-443, 2011.
- J.J. Manfredi, M. Parviainen, and J.D. Rossi. On the definition and properties of p-harmonious functions. Ann. Sc. Norm. Super. Pisa Cl. Sci.11(2):215-241, 2012.
- P. Juutinen, E. Saksman. Hamilton-Jacobi flows and characterization of solutions of Aronsson equations. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5). 6: 1-13, 2007.
- J. Heinonen, T. Kilpeläinen, and O. Martio. Nonlinear potential theory of degenerate elliptic equations. Dover Publications. 2006.
- G. Aronsson, M. G. Crandall, and P. Juutinen. A tour of the theory of absolutely minimizing functions. Bull. Amer. Math. Soc. 41: 439-505, 2004.
- T. Kilpeläinen, J. Malý. The Wiener test and potential estimates for quasilinear elliptic equations, Acta Math. 172: 137- 161, 1994.

## Collaboration

- Geometric Analysis group, University of Jyväskylä
- Inverse problems group, University of Jyväskylä
- Stochastics group, University of Jyväskylä
- Nonlinear PDE group, Aalto University
- Marco Barchiesi, University of Naples
- Anders Björn, Linköping University
- Jana Björn, Linköping University
- Verena Bögelein, University of Salzburg
- Frank Duzaar, Universität Erlangen-Nürnberg
- Martin Fuchs, Universität des Saarlandes
- Nicola Fusco, University of Naples
- Ugo Gianazza, University of Pavia
- Bernd Kawohl, University of Cologne
- Juha Kinnunen, Aalto University
- Tuomo Kuusi, Aalto University
- Peter Lindqvist, NTNU Trondheim
- Jan Malý, Charles University in Prague
- Juan J. Manfredi, University of Pittsburgh
- Olli Martio, University of Helsinki
- Giuseppe Mingione, University of Parma
- Kaj Nyström, Uppsala University
- Julio D. Rossi, University of Buenos Aires
- Eero Saksman, University of Helsinki
- Xiao Zhong, University of Helsinki

## Seminars

Analysis seminar and seminar on stochastics and PDEs at University of Jyväskylä.