Classical Potential Theory

Our starting point is Laplace’s equation Δ h= 0 on an open subset Ω of ℝ ^N , where \(\Delta = {\partial ^2}/\partial x_1^2 + \cdots + {\partial ^2}/\partial x_N^2.\)

We start with an algebraic study of harmonic polynomials as elements of vector spaces equipped with inner products. This leads quickly to information about the structure of the spaces and the behaviour of individual elements. The special role of axially symmetric polynomials is emphasized. We next extend the study of polynomial expansions of harmonic functions which concluded Chapter 1 and give an expansion for harmonic functions on annular domains, analogous to the Laurent expansion for holomorphic functions. This Laurent-type expansion is then used to obtain basic results on harmonic approximation. These results will be applied firstly to establish the existence of harmonic functions with prescribed singular parts at a sequence of isolated singularities, and secondly to construct harmonic functions on ℝ ^N with unexpected properties.

We have seen that harmonic functions on an open set Ω can be characterized as those finite-valued, continuous functions h on Ω which satisfy the mean value property: h (x) = M (h;x,r) whenever \( \overline {B\left( {x,r} \right)} \subset \Omega \). Subharmonic functions correspond to one half of this definition — they are upper-finite, upper semicontinuous functionss which satisfy the mean value inequality s (x) ≤ M (s;x,r) whenever \( \overline {B\left( {x,r} \right)} \subset \Omega \). They are allowed to take the value −∞ 00 so that we can include such fundamental examples as \( \log \left\| x \right\|\left( {N = 2} \right)\) and \( - {\left\| x \right\|^{2 - N}}\left( {N \geqslant 3} \right)\). Also, semicontinuity (rather th an continuity) is the appropriate condition for certain key results (for example, Theorems 3.1.4 and 3.3.1) to hold. The reason for the name “subharmonic” will become apparent in Section 3.2.

We recall that, if y ∈ ℝ ^N , then the function defined by is superharmonic on ∝ ^N and harmonic on ∝ ^N \{y}

Sets on which a superharmonic function can have the value +∞ are called polar. Since superharmonic functions are locally integrable, such sets must be of Lebesgue measure zero. Indeed, polar sets are the negligible sets of potential theory and will be seen to play a role reminiscent of that played by sets of measure zero in integration. A useful result proved in Section 5.2 is that closed polar sets are removable singularities for lower-bounded superharmonic functions and for bounded harmonic functions. In Section 5.3 we will introduce the notion of reduced functions. Given a positive superharmonic function u on a Greenian open set Ω and E ⊆ Ω, we consider the collection of all non-negative superharmonic functions v on ∖ which satisfy v ≥ u on E. The infimum of this collection is called the reduced function of u relative to E in Ω. Some basic properties of reduced functions will be observed, including the fact that they are “almost” superharmonic. Later, in Section 5.7, deeper properties will be proved via an important result known as the fundamental convergence theorem of potential theory. Before that, however, we will develop the notion of the capacity of a set, beginning with compact sets. Taking u ≡ 1 and E to be compact, the above reduced function is almost everywhere equal to a potential on Ω, and the total mass of the associated Riesz measure is called the capacity of E. For arbitrary sets E, we will define inner and outer capacity and, if these are equal, will term E capacitable.

In its simplest form the Dirichlet problem may be stated as follows: for a given function \( f \in C\left( {{\partial ^\infty }\Omega } \right)\), determine, if possible, a function h ∈ H(Ω) such that h(x) → f(y) as x → y for each \( y \in {\partial ^\infty }\Omega \). Such a function h is called the (classical) solution of the Dirichlet problem on Ω with boundary function f, and the maximum principle guarantees the uniqueness of the solution if it exists. For example, if Ω is either a ball or a half-space and f ∈ C(δ^∞Ω), then the solution of the Dirichlet problem certainly exists and is given by the Poisson integral of f. This follows immediately from Theorems 1.3.3 and 1.7.5. On the other hand, there are quite simple examples in which there is no such solution.

Let (x _n ) be a sequence of points in B\{0} converging to 0, and let

We saw in Chapter 1 that if μ is a measure on S, then the equation where K is the Poisson kernel of B, defines a non-negative harmonic function h on B, and that every such function h has a unique representation of this form. In a more general domain Ω non-negative harmonic functions need not have such a representation involving measures on δ^∞Ω.

We saw in Chapter 4 that a potential on the unit ball B has radial limit 0 at σ-almost every boundary point (Littlewood’s theorem), and that a positive harmonic function on B has finite non-tangential limits at σ-almost every boundary point (Fatou’s theorem). The notions of radial and non-tangential limits are clearly unsuitable for the study of boundary behaviour in general domains. To overcome this difficulty, we will develop the ideas of the preceding two chapters by defining the “minimal fine topology” on the Martin compactification, \( \hat \Omega \), of a Greenian domain Ω. (When restricted to Ω, this topology reduces to the fine topology we have already studied.) Thus we will be able to discuss minimal fine limits of functions at (minimal) Martin boundary points and the concept of a set being “minimally thin” at such points.