In this section we want to construct solutions of the classical Dirichlet prob-lem similar to methods known from complex analysis, where the value f(z) of a given holomorphic function f is already determined by its values on the boundary of an arbitrary disk. More precisely.
Theorem 1.3.1 (Cauchy’s integral formula) Let Ω⊆C be a (complex) domain and f: Ω−→C be holomorphic. If B(z, r)⊆Ω
∀z ∈B(z, r) f(z) = 1 2πi
I
|ζ−z|=r
f(ζ) ζ−z dζ ,
where the parameterization around the boundary is counter-clockwise.
Note that the valuef(z) is given by integration over the fundamental function ζ 7−→(ζ−z)−1. Before we can apply this concept to harmonic functions, we need some more basics, in particular, we have to define distributions.
Definition 1.3.2 Let C∞(Ω) denote functions in \
k∈N
Ck(Ω). Furthermore, suppf :={x∈Ω : f(x)6= 0} (support of f),
M bΩ :⇐⇒M ⊆Ω and M is compact,
C0k(Ω) :={f ∈ Ck(Ω) : suppf bΩ} fork ∈N∪ {∞} .
Functions in C0∞(Ω) are called test functions. A member ϕ ∈ C0∞(Ω) for B(0,1)⊆Ω is, e.g.,
ϕ(x) :=
(exp(−1/(1−x2)) if |x|<1 ,
0 otherwise.
By defining ϕε(x) :=ϕ(x/ε)·c/εn, wherecis such that R
Rnϕε(x)dx= 1, we obtainFriedrichs’ mollifier. Letf: M ⊆Rn−→Rbe Lebesgue measurable.
Definition 1.3.3 Lebesgue spaces Lp(M) for 1 ≤ p < ∞ are given for functions f: M −→R by means of the norm
|f|Lp(M) :=Z
M
|f(x)|p dx1/p
, and then Lp(M) :={f: M −→R : |f|Lp(M) <∞} ,
Lploc(M) :={f: M −→R : ∀M0 bM f ∈ Lp(M0)} .
Actually, we consider, as usual, quotient spaces of these Lebesgue spaces, i.e., modulo functions of Lebesgue measure zero (but we use the same symbols).
If an equivalence class contains, e.g., a continuous function, we may also implicitly identify the class with this function. Note that L2(M) is a Hilbert space with respect to the inner product
(f, g)L2(M):=
Z
M
f(x)g(x) dx ,
while for 1 ≤p <∞ all Lp(M) are Banach spaces with respect to | · |Lp(M). Also note that C0∞(Ω) is dense in L2(Ω) with respect to L2-norm. Let D(Ω) :=C0∞(Ω) be endowed with the following concept of convergence.
Definition 1.3.4 A sequenceϕj ∈ C0∞(Ω)is called aD-zero sequenceif there isM bΩwith∀j ∈N suppϕj ⊆M and ∀β ∈Nn supx∈Ω|∂βϕj(x)| −→
j→∞0. Definition 1.3.5 A linear functional D(Ω) −→R is called D-continuous if D-zero sequences are mapped to R-zero sequences. Let D0(Ω) be the space of D-continuous linear functionals. Its elements are called distributions.
Each distribution T ∈ D0(Ω) for which there exists a function f ∈ L1loc(Ω) withT(ϕ) =R
Ωf(x)ϕ(x)dxfor all test functionsϕ∈ C0∞(Ω) is calledregular. We may identify a function f ∈ L1loc(Ω) with the corresponding distribution.
A non-regular distribution is, e.g., the Dirac distribution δ.
Definition 1.3.6 Let 0 ∈ Ω. The Dirac distribution δ: C0∞(Ω) −→ R is defined by δ(ϕ) :=ϕ(0).
We briefly show that δ is not regular. Hence, we cannot avoid the use of distributions for Def. 1.3.7.
Proof: Assume there exists f ∈ L1loc(Ω) with
∀ϕ∈ C0∞(Ω) δ(ϕ) = Z
x∈Ω
f(x)ϕ(x) dx . Let ε > 0 with R
|x|<ε|f(x)| dx < 1 (and B(0, ε) ⊆ Ω), and let ϕε ∈ C0∞(Ω) be Friedrich’s mollifier. Then we obtain the contradiction
ϕε(0) = Z
x∈Ω
f(x)ϕε(x)dx≤ϕε(0) Z
|x|<ε
|f(x)| dx < ϕε(0).
Now we can define fundamental solutions.
Definition 1.3.7 A function f ∈ L1loc(Rn) is called fundamental solution in Rn if −∆f =δ in the sense of distributions.
In fact, we have already implicitly used fundamental solutions for the proof of the meanvalue property, Lemma 1.2.7, and in Sect.1.1, we already saw a fundamental solution, a constant times |x|−1, for R3.
Theorem 1.3.8 The function is a fundamental solution in Rn.
Proof: Clearly, hn ∈ L1loc(Rn). Because we will apply the Green identity, Theorem 1.2.6, note that hn is harmonic on Rn \ {0}, which we already showed for n ≥ 3 in (1.11), but also holds for n = 1,2 with the same rhs
where (∗) is an application of the Green identity onB(0, ε, r), andris chosen, such that the support of ϕ is contained. The notationO(ε) here denotes the asymptotical behavior for ε→0. In the last integral, the term∂nhn is equal to ε1−n/ωn. Hence, we obtain Now we can state the first main theorem of this section, using the fundamen-tal solution hn. where ∂nξ is the normal derivative with respect to ξ.
Proof: Let ϕ ∈ C0∞(Ω) with ϕ = 1 in a neighborhood of x ∈Ω. Integration Unfortunately, apart from the values of f on ∂Ω, also its normal derivatives are involved. Thus, in general, i.e., for arbitrary ∂Ω ∈ C2,α, the classical Dirichlet problem (where only Dirichlet boundary conditions are given, and normal derivatives of f on the boundary are unknown) is not yet solved by Theorem 1.3.9. (Anyhow, in the next chapter we will see how the solution can be approximated by means of the Laplacian matrix.) However, for spe-cial shapes of Ω, the solution can explicitly be given by means of Green’s functions.
Definition 1.3.10 A function Γ : Ω×Ω−→R is called Green’s function if
• ∀x∈Ω Γ(·, x)∈ C(Ω\ {x})∩ C2(Ω\ {x}) and −∆Γ(·, x) =δ(· −x),
• ∀x∈Ω ∀ξ ∈∂Ω Γ(ξ, x) = 0 .
A Green’s function Γ(·, x) is to replace the fundamental solution hn(· −x) in the formula of Theorem 1.3.9. The first requirements on Γ state that this is possible within Ω. The second requirement assures that Γ(·, x) vanishes on the boundary ∂Ω. Hence, in the formula of Theorem 1.3.9, the normal derivatives of the solution f are no more needed, and the solution can be given just in terms of its values on the boundary ∂Ω. For special shapes of Ω, a Green’s function can be given directly. Let Ω := B(0, r) and x∈Ω
where Kr denotes the Kelvin transform (see (1.7)), defines a Green’s func-tion. The idea is to use the difference of the fundamental solutionhninside Ω and the Kelvin transform of hn outside Ω. If we apply this to the formula of Theorem 1.3.9, we obtain the celebrated Poisson formula which completely solves the classical Dirichlet problem on balls. This is the second main the-orem of this section.
is the solution of the classical Dirichlet problem (Ω, g).
The proof consists of determining the normal derivatives of Γn(·, x) on the boundary ∂Ω. While also for other special shapes of Ω, a Green’s function can be given explicitly, e.g., by using conformal mappings2 to map harmonic functions onto other domains, in general, a direct computation is not easy.
Finally, we discuss the question for what kind of shapes of Ω the solution of the classical Dirichlet problem exists. In this section we just state the result that will be proved in Sect. 1.5. Recall the meanvalue property for harmonic functions. Now we define subharmonic (superharmonic) functions.
Definition 1.3.12 A function f ∈ C(Ω) is called subharmonic (superhar-monic) if for all x∈Ω ∀r >0 with B(x, r)⊆Ω
Definition 1.3.13 A function b(·, x)∈ C(Ω) is called a barrier function for x ∈∂Ω if b(·, x) is superharmonic, b(x, x) = 0 and positive everywhere else.
A point x∈∂Ω is called regular if there exists a barrier function for x.
A sufficient condition for regularity of a point x∈∂Ω is as follows.
2The Kelvin transform in Rn is a conformal mapping. InR2, where potential theory is developed very well because many results from complex analysis can be used (e.g., real and imaginary part of a holomorphic function correspond to a harmonic function), every holomorphic function, one-to-one onC∪ {∞} −→C∪ {∞}, yields a conformal mapping.
Lemma 1.3.14 The point x ∈ ∂Ω is regular if there exist ε > 0, x0 ∈ Rn, such that B(x0, ε)∩Ω =∅ and B(x0, ε)∩Ω ={x}.
Proof: Forn ≥3, the functionb(·, x) :=ε2−n−|·−x0|2−nis a barrier function.
Forn = 2, useb(·, x) := ln|·−x0|−lnε, and forn= 1, useb(·, x) := |·−x0|−ε
as a barrier function.
One can proof the same result for an open cone instead of a ball.
The following concluding main theorem of this section characterizes the shapes of Ω for which the classical Dirichlet problem is solvable. A proof is given in Sect. 1.5.
Theorem 1.3.15 A classical Dirichlet problem (Ω, g)is solvable if and only if each x∈∂Ω is regular.