(in Polish) Wstęp do metod lepkościowych i potencjalnych dla wszelkich równań różniczkowych cząstkowych 1000-1S20WML
We wish to present a thread of development of PDE's based on Perron's method for the Laplace operator. One line of development is devoted to the method of solving linear on nonlinear problems with the help of the vanishing viscosity. Another approach deals with methods for studying regularity of solutions of nonlinear problems. It is based on the analysis of certain potentials (i.e. Wolff potentials) for measures generated by solutions obtained with the Perron method.
The idea at the heart of the vanishing viscosity method is the introduction of an additional term guaranteeing existence of the approximate problem. Its convergence is obtained due to the comparison principle using the maximum principle.
The maximum principle is also at the core of the Perron's method.
We will present the Perron method in the classical setting for the Laplace equation. We will talk about the vanishing viscosity method for the Hamilton-Jacobi equations. We will introduce the notion of viscosity solution for elliptic, parabolic and special first order problems. We will also deal with the current trends of using the viscosity theory for integro-differential problems.
We will explain the success of the viscosity theory. Namely, with its help one may construct solutions there, where other methods fail due to appearance of singularities. A well-known example is the geometric problem of the mean curvature flow. We will present it according to the taste of the participants.
Perron's method is also at the heart of a method for studying regularity of solutions. It is based on the fact that the classical notions of the super- and sub-harmonic function related to the Laplace operator could be generalized for a broad class of nonlinear operators. So-called A-super harmonic functions are neither super- nor sub-solutions in a strict sense. They rather satisfy a comparison principle for weak solutions of nonlinear homogeneous problems, i.e. they are a generalization of a harmonic function. It turns out that the differential operator acting on an A-harmonic function generates a measure. The analysis of behavior of so-called Wolff potential for this measure permits a regularity (i.e. regularity, differentiability) study of generalized solutions of non-homogeneous nonlinear elliptic problems.
We will present this method in the case of the p-Laplace operator and its generalizations also in the vectorial case, i.e. for systems.
Main fields of studies for MISMaP
Type of course
Mode
Learning outcomes
A student:
1. knows the notion of viscosity solutions for equation admitting the comparison principle
2. knows the Perron method.
3. knows how to apply these notions.
Assessment criteria
In order to get credit a student must present at least on talk per semester.
Bibliography
[BCCI] G.Barles, E.Chasseigne, A.Ciomaga, C.Imbert, Lipschitz regularity of solutions formixed integro-differential equations, J. Differential Equations, 252 (2012), 6012-6060.
[C] M.G. Crandal, Viscosity solutions: a primer, Lecture Notes in Math., 1660, Fond. CIME/CIME Found. Subser., Springer, Berlin, 1997
[CIL] M.G.Crandall, H.Ishii, P.-L.Lions, User's guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. (N.S.) 27 (1992), no. 1, 1-67
[GT] D. Gilbarg, N.S. Trudinger, Elliptic partial differential equations of second order, Springer, Berlin, 1983.
[I] H.Ishii, Perron's method for Hamilton-Jacobi equations. Duke Math. J. 55 (1987), no. 2, 369-384.
[HKM] J. Heinonen, T. Kilpelainen, O. Martio, Nonlinear potential
theory of degenerated elliptic equations, Courier Dover Publications,
2006;
[KM] T. Kilpelainen, J. Maly, Degenerate elliptic equations with
measure data and nonlinear potentials, Annali della Scuola Normale
Superiore di Pisa, 1992
[KuMi] T. Kuusi, G. Mingione, Vectorial nonlinear potential theory,
J. Europ. Math. Soc. (JEMS) (2018)
Additional information
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