Calculus of Variations and Geometric Measure Theory

S. J. N. Mosconi

Discrete regularity for elliptic equations on graphs

created on 28 May 2001
modified on 18 Apr 2003


Submitted Paper

Inserted: 28 may 2001
Last Updated: 18 apr 2003

Pages: 20
Year: 2001


The paper deals with the regularity properties of solutions to elliptic difference equation on a graph $G$ under suitable hypotheses on the geometry of the latter, namely a doubling condition and a Poincarè inequality involving discrete derivatives. The difference equation we deal with is of the kind $\sum_{x\sim y}a(x,y)(u(y)-u(x)) + c(x)u(x)=f(x)$ where $x\sim y$ means that the vertices $x$ and $y$ of $G$ are connected each other in the graph structure. The main hypotheses on the coefficients of the equation is that $a$ is symmetric in its arguments and positive. A Harnack inequality for positive solutions is proved, thus giving an Hölder estimate on the oscillation of a solution in term of suitable $L^p$ norms of the coefficients.

Keywords: graph, elliptic, Harnack, regularity