Harnack inequalities for integro-differential operators

Authors

Mohammud Foondun, University of Connecticut

Date of Completion

January 2006

Keywords

Mathematics

Degree

Ph.D.

Abstract

In the first part of this dissertation, we consider the operator L defined on C² ( R^d ) functions by Lfx =12 i,j=1

d

aijx 6²fx 6xi6xj +i=1

d

bi x6fx 6xi +R ^d\0 fx+h -fx-1 h≤1h˙1 fxn x,hdh. Under the assumption that the local part of the operator is uniformly elliptic and with suitable conditions on n( x, h), we establish a Harnack inequality for functions that are nonnegative and harmonic in a domain. We also show that the Harnack inequality can fail without suitable conditions on n(x, h). A regularity theorem for those nonnegative harmonic functions is also proved. ^ In the second part, we consider the Dirichlet form given by Ef,f =12 R^d i,j=1

d

aijx 6fx 6xi6f x6xj dx +R^d ×R^d fy-fx ²Jx,y dxdy. Under the assumption that the {a_ij } are symmetric and uniformly elliptic and with suitable conditions on the nonlocal part, we obtain upper and lower bounds on the heat kernel of the Dirichiet form. We also prove a Harnack inequality and a regularity theorem for functions that are harmonic with respect to E .^

Recommended Citation

Foondun, Mohammud, "Harnack inequalities for integro-differential operators" (2006). Doctoral Dissertations. AAI3234304.
https://digitalcommons.lib.uconn.edu/dissertations/AAI3234304