Harmonic Functions

Abstract

This part begins with a review of basic properties of harmonic functions, whose consideration is suggested by several branches of physics (Newtonian gravity, electrostatics) and mathematics (e.g. complex analysis and the theory of surfaces). When the theory of surfaces was developed, the Laplace-Beltrami operator was viewed as a differential parameter of order 2, and this is shown in detail. The reader is then introduced to the theory of distributions and Sobolev spaces, while regularity theory is introduced through the Caccioppoli derivation of integral bounds on solutions of linear elliptic equations. The concept of ellipticity is then defined in various cases of interest. After an outline of spectral theory, the De Giorgi example of Laplace equation with mixed boundary conditions is studied. As a next step, Morrey and Campanato spaces, and functions of bounded mean oscillation, are studied. Part II ends by focusing on pseudo-analytic functions, with the associated generalized form of Cauchy-Riemann systems, and some remarkable properties of biharmonic and polyharmonic functions.

Appendix 3.a: Frontier of a Set and Manifolds with Boundary

Since the concept of boundary plays an important role in this section, we consider it in greater detail in this appendix.

Given a subset E of $\mathbf{R}^{n}$, it closure is denoted by ${\overline{E}}$, the interior by $E^{\mathrm{o}}$ and the frontier by

$$\begin{aligned} \partial E \equiv {{\overline{E}} \over E^\mathrm{o}}. \end{aligned}$$

(3.a1)

A richer concept is the one of manifold with boundary , a prototype of which is the open ball of Euclidean space, whose boundary is the corresponding sphere as we have seen in (3.4.15). Following Schwartz [125], a manifold with boundary, of class $C^{m}$ and dimension n, is a part V of a manifold ${\widetilde{V}}$ of class $C^{m}$ and dimension n, bounded, identical with the closure of its interior:

$$\begin{aligned} V={\overline{V^{\mathrm{o}}}}, \end{aligned}$$

(3.a2)

whose frontier ${\partial V}=\Sigma$ is a hypersurface of ${\widetilde{V}}$, sub-manifold of class $C^{m}$ and dimension $n-1$. This frontier is then said to be the boundary of V, also denoted by bV.

Some of the topological properties which hold when V is a ball of an Euclidean space, and bV the corresponding sphere, can be extended to the general case. If $\Sigma$ is empty, then V is simply an ordinary manifold (also called without boundary or closed). Let us therefore consider the case where $\Sigma$ is not empty. Its complementary set $C \Sigma$ within ${\widetilde{V}}$ is the union of two disjoint open sets, i.e. the interior $V^{\mathrm{o}}$ of V, and the complementary set CV, exterior to V. None of these two open sets can be empty.

Indeed, if $V^{\mathrm{o}}$ were empty, then by virtue of (3.a2) V would be necessarily empty as well, being the closure of an empty set. On the other hand, if CV were empty, then V would be identical with ${\widetilde{V}}$, and hence its frontier $\Sigma$ would be empty. It then follows that $C \Sigma$, being the union of two disjoint and non-empty open sets, is certainly not connected. It contains at least two regions. If we denote by $\left( \Omega _{i}\right) _{i \in I}$ the connected components or regions of $C \Sigma$, each of them, if it has at the same time points in common with $V^{\mathrm{o}}$ and with CV, must have points in common with $\Sigma$, which is instead impossible because it lies in the complementary set of $\Sigma$; thus, each of the regions $\Omega _{i}$ lies entirely in $V^{\mathrm{o}}$ or entirely in CV. In other words, one can obtain a partition of the set I in the form of union of two complementary sets J and K, and one has

$$\begin{aligned} V^{\mathrm{o}}=\cup _{i \in J}(\Omega _{i}), \; CV=\cup _{i \in K}\Omega _{i}. \end{aligned}$$

(3.a3)

The interior and the exterior of V are unions of regions of the complementary set of $\Sigma$. If one finds that $\Sigma$ splits V exactly into two regions, then these two regions are necessarily $V^{\mathrm{o}}$ and CV.