Illinois Journal of Mathematics
- Illinois J. Math.
- Volume 45, Number 3 (2001), 1021-1050.
Harmonic functions on metric spaces
This paper explores a Dirichlet type problem on metric measure spaces. The problem is to find a Sobolev-type function that minimizes the energy integral within a class of "Sobolev" functions that agree with the boundary function outside the domain of the problem. This is the analogue of the Euler-Lagrange formulation in the classical Dirichlet problem. It is shown that, under certain geometric constraints on the measure imposed on the metric space, such a solution exists. Under the condition that the space has many rectifiable curves, the solution is unique and satisfies the weak maximum principle.
Illinois J. Math., Volume 45, Number 3 (2001), 1021-1050.
First available in Project Euclid: 13 November 2009
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 31C45: Other generalizations (nonlinear potential theory, etc.)
Secondary: 30C65: Quasiconformal mappings in $R^n$ , other generalizations 49J40: Variational methods including variational inequalities [See also 47J20]
Shanmugalingam, Nageswari. Harmonic functions on metric spaces. Illinois J. Math. 45 (2001), no. 3, 1021--1050. doi:10.1215/ijm/1258138166. https://projecteuclid.org/euclid.ijm/1258138166