Illinois Journal of Mathematics

Harmonic functions on metric spaces

Nageswari Shanmugalingam

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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.

Article information

Illinois J. Math., Volume 45, Number 3 (2001), 1021-1050.

First available in Project Euclid: 13 November 2009

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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.

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