Spectral theory for the fractal Laplacian
Hans Triebel · Birkhäuser Basel eBooks · 2001
We modify the explanations given in [Triδ], 30.1, pp. 233-234. LetΩbe a bounded domain in the plane $$ {\mathbb{R}^2} $$ withC ∞ boundary ∂Ω, interpreted as a membrane fixed at its boundary. Vibrations of such a membrane in $$ {\mathbb{R}^3} $$ are measured by the deflection v(xt), where x = (x 1 ,x 2 )∈Ω, and t≥ 0 stands for the time. In other words, the point (x1, x2, 0) in $$ {\mathbb{R}^3} $$ with (x1, x2) ∈Ωof the membrane at rest, is deflected to (x1, x2, v(x,t)). Up to constants the usual physical description is given by 19.1 $$ \Delta v\left( {x,t} \right) = m\left( x \right)\frac{{{\partial ^2}v\left( {x,t} \right)}}{{\partial {t^2}}},x \in \Omega ,t \geqslant 0, $$ and 19.2 $$ v\left( {y,t} \right) = 0 if y \in \partial \Omega ,t \geqslant 0, $$ where $$ \Delta = \frac{{{\partial ^2}}}{{\partial x_1^2}} + \frac{{{\partial ^2}}}{{\partial x_2^2}} $$ and the right-hand side of (19.1) is Newton’s law with the mass densitym(x).To find the eigenfrequencies one has to insert v(x,t) = u(x)e iλt with λ∈ $$ \mathbb{R} $$ in (19.1) and obtains 19.3 $$ - \Delta u\left( x \right) = {\lambda ^2}m\left( x \right)u\left( x \right),x \in \Omega ;u\left( y \right) = 0 if y \in \partial \Omega, $$ where one is interested in non-trivial solutionsu(x).Hence one asks for eigen-functions and eigenvalues of the operator 19.4 $$ B = {\left( { - \Delta } \right)^{ - 1}}^\circ m\left( \cdot \right), $$ where (−Δ)−1 is the inverse of the Dirichlet Laplacian −Δ.We use the notationDirichlet Laplacianalways with the understanding that vanishing boundary data at óS2 are incorporated into the domains of definition for −Δ in the function spaces considered, preferably $$ B_{pq}^s\left( \Omega \right) $$ and $$ H_{pq}^s\left( \Omega \right) $$ with 1 \frac{1}{p} $$ (this will be specified in greater detail in the next subsection). If ϱ is a positive eigenvalue of B then $$\lambda = \varrho ^{ - \frac{1} {2}} $$ is the related eigenfrequency. We are interested in the problem of what happens when the mass densitym(x)shrinks to a fractal set Γand a related Radon measure µ with 19.5 $$ supp \mu = \Gamma \subset \Omega$$ .