Subharmonic function

Let $U\subseteq\mathbb R^m$ be open. A function $u:U\to[-\infty,\infty)$ , not identically $-\infty$ on any connected component, is subharmonic if it is upper-semicontinuous and

u(x)\le \frac{1}{|S^{m-1}|}\int_{S^{m-1}}u(x+r\omega)\,d\omega

whenever the closed ball $\overline{B_r(x)}$ lies in $U$ . For $m=2$ , after identifying $\mathbb R^2\cong\mathbb C$ , this is the usual circular sub-mean inequality.

Distributional characterization

For a function $u\in L^1_{\mathrm{loc}}(U)$ , the condition $\Delta u\ge0$ in the distributional sense is equivalent to the existence of a unique upper-semicontinuous subharmonic representative of the almost-everywhere class of $u$ . Thus the distributional condition characterizes the representative, rather than the values assigned to an arbitrary $L^1_{\mathrm{loc}}$ version. For example, the function that is $1$ at one point and $0$ elsewhere has $\Delta u=0$ as a distribution and is upper-semicontinuous, but fails the sub-mean inequality at that point. For $u\in C^2$ , this becomes the pointwise inequality $\Delta u\ge0$ .

Examples

Harmonic functions satisfy equality in the mean-value formula and are subharmonic. Every convex function on a convex open set is subharmonic. If $f$ is holomorphic and not identically zero on any connected component, then $\log|f|$ is subharmonic, with its distributional Laplacian recording the zeros of $f$ . The qualification is necessary: for $f\equiv0$ , $\log|f|\equiv-\infty$ , which is excluded by the definition on that component.

References

Thomas Ransford, Potential Theory in the Complex Plane, Cambridge University Press, 1995. DOI record.
David H. Armitage and Stephen J. Gardiner, Classical Potential Theory, Springer, 2001. DOI record.