Illinois Journal of Mathematics

Local compactness for families of {$\scr A$}-harmonic functions

K. Rogovin

Full-text: Open access


We show that if a family of $\mathcal{A}$-harmonic functions that admits a common growth condition is closed in $L^p_{\operatorname{loc}}$, then this family is locally compact on a dense open set under a family of topologies, all generated by norms. This implies that when this family of functions is a vector space, then such a vector space of $\mathcal{A}$-harmonic functions is finite dimensional if and only if it is closed in $L^p_{\operatorname{loc}}$. We then apply our theorem to the family of all $p$-harmonic functions on the plane with polynomial growth at most $d$ to show that this family is essentially small.

Article information

Illinois J. Math., Volume 48, Number 1 (2004), 71-87.

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: 30C62: Quasiconformal mappings in the plane 30C65: Quasiconformal mappings in $R^n$ , other generalizations 35J60: Nonlinear elliptic equations


Rogovin, K. Local compactness for families of {$\scr A$}-harmonic functions. Illinois J. Math. 48 (2004), no. 1, 71--87. doi:10.1215/ijm/1258136174.

Export citation