The fractal Dirichlet problem

Abstract

Let Ω be a bounded C ^∞ domain in ${\mathbb{R}^n}$ where, temporarily, n ≥ 3, and let Γ=∂Ω be equipped naturally with a Radon measure μ equivalent (or equal) to ${\mathcal{H}^{n - 1}}\left| \Gamma \right.$ (the restriction of the Hausdorff measure ${\mathcal{H}^{n - 1}}$ in ${\mathbb{R}^n}$ to). The single layer potential G,

$$ \left( {Gh} \right)\left( x \right) = \int\limits_\Gamma {\frac{{h\left( r \right)}}{{{{\left| {x - \gamma } \right|}^{n - 2}}}}} \mu \left( {d\gamma } \right),x \in {\mathbb{R}^n}, $$

(20.1)

makes sense both in ${{\mathbb{R}}^n}$ and on Γ (using the same letterG) if, for example, his bounded. Since Γ is a compact C ^∞ manifold,

$$ {H^s}\left( \Gamma \right) = B_{2,2}^s\left( \Gamma \right),s \in \mathbb{R}, $$

(20.2)

can be introduced in a canonical way via local charts. It turns out thatG (restricted to Γ) makes sense for some spaces H ^s (Γ). In particular,

$$ G{H^{ - \frac{1}{2}}}\left( \Gamma \right) = {H^{\frac{1}{2}}}\left( \Gamma \right) $$

(20.3)

is an isomorphic mapping. This has the consequence that the uniquely determined solution u(x) of the (almost) classical Dirichlet problem

$$ \Delta u\left( x \right) = 0,x \in \Omega,u \in {H^1}\left( \Omega \right), $$

(20.4)

$$ t{r_\Gamma }u = g \in {H^{\frac{1}{2}}}\left( \Gamma \right), $$

(20.5)

for given g, can be uniquely represented by (20.1) as u =Ghwith some $h \in {H^{ - \frac{1}{2}}}\left( \Gamma \right)$ . It is the main aim of this section to extend these observations to fractals. We rely on the techniques developed in Section19. We describe what can be expected. First we remark that the boundary Γ=∂Ω of the above C ^∞ domain Ω is an (n − 1)-set according to (18.1). In particular, the singularity |x − γ|²⁻ⁿ with x ∈ Γ and γ ∈ Γ, is well compensated by (18.1) with d = n ≤ 1.