36 research outputs found
On the Rough Gronwall lemma and it's aplications
No full textHofmanová M. On the Rough Gronwall lemma and it's aplications. In: Eberle A, Grothaus M, Hoh W, Kassmann M, Stannat W, Trutnau G, eds. Stochastic Partial Differential Equations and Related Fields. In Honor of Michael Röckner SPDERF, Bielefeld, Germany. Springer Proceedings in Mathematics & Statistics . Cham: Springer; 2016: 333-344
Harnack Inequalities: An Introduction
No full textThe aim of this article is to give an introduction to certain inequalities named after Carl Gustav Axel von Harnack. These inequalities were originally defined for harmonic functions in the plane and much later became an important tool in the general theory of harmonic functions and partial differential equations. We restrict ourselves mainly to the analytic perspective but comment on the geometric and probabilistic significance of Harnack inequalities. Our focus is on classical results rather than latest developments. We give many references to this topic but emphasize that neither the mathematical story of Harnack inequalities nor the list of references given here is complete
The theory of De Giorgi for non-local operators
No full textKaßmann M. The theory of De Giorgi for non-local operators. Comptes Rendus Mathematique. 2007;345(11):621-624
The Cauchy problem and the martingale problem for integro-differential operators with non-smooth kernels
No full textExistence of a Generalized Green Function for Integro-Differential Operators of Fractional Order
No full textRobust nonlocal trace and extension theorems
No full textWe prove trace and extension results for Sobolev-type function spaces that
are well suited for nonlocal Dirichlet and Neumann problems including those for
the fractional -Laplacian. Our results are robust with respect to the order
of differentiability. In this sense they are in align with the classical trace
and extension theorems.Comment: In the updated version we fixed some irritating typo
Nonlocal operators related to nonsymmetric forms II: Harnack inequalities
No full textLocal boundedness and Harnack inequalities are studied for solutions to parabolic and elliptic integro-differential equations whose governing nonlocal operators are associated with nonsymmetric forms. We present two independent proofs, one being based on the De Giorgi iteration and the other one on the Moser iteration technique. This article is a continuation of a recent work by the same authors, where Hölder regularity and a weak Harnack inequality are proved in a similar setup
