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Harnack Inequalities for Stochastic Partial Differential Equations

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Nonlinear Partial Differential Equations — Department of Mathematics and Statistics

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21. Stochastic Differential Equations

Unavailable for purchase. Continue shopping Checkout Continue shopping. 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. 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.


  • Stochastic Partial Differential Equations and Related Fields.
  • Nonlinear Partial Differential Equations — Department of Mathematics and Statistics.
  • Bibliographic Information.

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. 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. 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. 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. 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.

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There are three types of natural questions under research:. 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. The most up-to-date information is available on the members' personal web pages.


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Nonlinear Partial Differential Equations. Shape Optimization 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. Parabolic PDEs 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.

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? Some publications A. Attouchi, M. Parviainen, E. Pures Appl. Barchiesi, Marco, A.