25.1.2 Harmonic functions and boundary values16FFFBC63146

Fundamental solutions, the regularity of distributional solutions of the Laplace equation, the Poisson formula and Harnack's inequality are treated in [Theorems 3.1.2–3.1.7, pp. 117–123, Hor07]. As there, the fundamental solution $E$ of proposition 3.1.13 satisfies $\Delta E=\delta _0$, whereas our Green functions are positive and satisfy $-\Delta G=\delta _ x$. Our proof of Weyl's lemma, theorem 3.1.11, uses radial averaging and convolution; Theorem 3.1.3 of that book contains in addition a statement on analytic continuation. The Poisson formula and Harnack's inequality (theorem 3.1.18, proposition 3.1.19) are stated with the explicit Poisson kernel and its two-sided bounds, and they include the case of dimension one.

Perron's method, barriers and the solution of the Dirichlet problem for continuous boundary data are developed in [Section 3.3, pp. 171–174, especially Theorem 3.3.3, Definition 3.3.4 and Theorem 3.3.5, Hor07]. In theorem 3.1.25, theorem 3.1.27 we work with continuous subsolutions. For the characterization of regular boundary points by barriers compare also [Theorem 3.4.17, pp. 219–220, Hor07]. Our local barrier criterion, proposition 3.1.30, uses a positive continuous supersolution near the given boundary point whose continuous extension to the nearby boundary vanishes only at that point; the theorem in Hörmander's book allows more general exceptional sets. The Green function of theorem 3.1.34 is constructed only on bounded regular domains; within this class the theorem also shows that Green functions increase with the domain and converge along increasing sequences of domains whose union is again bounded and regular.