Nonlocal nonlinear potential theory and fractional integral operators

Janne Korvenpää

Research output: ThesisDoctoral ThesisCollection of Articles

Abstract

This thesis develops Potential Theory for nonlinear fractional Laplace type equations. These equations are nonlocal integro-differential equations defined as singular integrals. We study weak solutions and weak supersolutions of the equations, demonstrating that they behave as in the case of standard elliptic partial differential equations. We also define a related notion of superharmonic functions via a comparison with weak solutions. The superharmonic functions are used to give a nonlocal version of Perron's method for solving Dirichlet problems with general boundary data. To obtain all the required properties of superharmonic functions, we use a related obstacle problem as a tool. For this, several regularity results for the solution to the obstacle problem are proved. In addition, we study a notion of viscosity solutions to the considered equations. The results reveal that the classes of viscosity supersolutions and superharmonic functions are the same, and for bounded solutions, they coincide with the class of weak supersolutions. The thesis also studies the regularity of maximal functions by extending the regularity results of a fractional maximal operator to its local counterpart. Finally, we consider finitely randomized dyadic systems on metric measure spaces and apply them to functions of bounded mean oscillation.
Translated title of the contributionEpälokaali epälineaarinen potentiaaliteoria ja fraktionaaliset integraalioperaattorit
Original languageEnglish
QualificationDoctor's degree
Awarding Institution
  • Aalto University
Supervisors/Advisors
  • Kinnunen, Juha, Supervising Professor
  • Kinnunen, Juha, Thesis Advisor
  • Kuusi, Tuomo, Thesis Advisor
Publisher
Print ISBNs978-952-60-7077-3
Electronic ISBNs978-952-60-7078-0
Publication statusPublished - 2016
MoE publication typeG5 Doctoral dissertation (article)

Keywords

  • fractional Laplace equation
  • nonlocal operator
  • potential theory
  • superharmonic function
  • Perron's method
  • obstacle problem

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