Logarithmic convexity type continuous dependence results for discrete harmonic functions defined as solutions of the standard $C^0$ piecewise-linear approximation to Laplace's equation are proved.

This leads to the solution of equations of the form $$\int_S g (Q) \log|P - Q|dS (Q) = h (P), \quad P \in S.$$ This equation is reformulated using a special change of variable, leading to a new ...