Illinois Journal of Mathematics

Harmonic functions on metric spaces

Nageswari Shanmugalingam

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Abstract

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

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

Dates
First available in Project Euclid: 13 November 2009

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1258138166

Digital Object Identifier
doi:10.1215/ijm/1258138166

Mathematical Reviews number (MathSciNet)
MR1879250

Zentralblatt MATH identifier
0989.31003

Subjects
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]

Citation

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


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