Heat kernel estimates and Harnack inequalities for some Dirichlet forms with non-local part

By Mohammud Foondun. 2009

60 citations recorded by the source corpus.

We consider the Dirichlet form given by $$ {\cal E}(f,f) = \frac{1}{2}\int_{R^d}\sum_{i,j=1}^d a_{ij}(x)\frac{\partial f(x)}{\partial x_i} \frac{\partial f(x)}{\partial x_j} dx$$ $$ + \int_{R^d \times R^d} (f(y)-f(x))^2J(x,y)dxdy.$$ Under the assumption that the ${a_{ij}}$ are symmetric and uniformly elliptic and with suitable conditions on $J$, the nonlocal part, we obtain upper and lower bounds on the heat kernel of the Dirichlet form. We also prove a Harnack inequality and a regularity theorem for functions that are harmonic with respect to $\cal E$.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/88da9f5527560cd6cb7d1db46358b48607dd0748

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Higher Hölder regularity for mixed local and nonlocal degenerate elliptic equations (2022, methodology): “We mention the work of Foondun [27], where a Harnack inequality and local Hölder continuity are established.”
On Harnack Inequality and Hölder Regularity for Isotropic Unimodal Lévy Processes (2013, background): “• In [9] it is assumed that, for any r < 1, there exist constants c and α such that ν(x−z) 6 cr−αν(y − z), for |x− y| < r and |x− z| > r.”
A Hong-Krahn-Szegö inequality for mixed local and nonlocal operators (2021, background): “The study of mixed local/nonlocal operators has been recently received an increasing level of attention, both in view of their intriguing mathematical structure, which combines the classical setting and the features typical of nonlocal operators in a framework that is not scale-invariant [40, 45, 46”
A priori Holder estimate, parabolic Harnack principle and heat kernel estimates for diffusions with jumps (2008, background): “3 are claimed in [Fo] under similar assumptions, however we have some difficulty to follow some of the arguments there.”
Parabolic Harnack inequality and heat kernel estimates for random walks with long range jumps (2007, unknown): “[Fo], where a similar condition is used.”
On the regularity theory for mixed local and nonlocal quasilinear elliptic equations (2021, methodology): “5, 24 [27] Mohammud Foondun.”