Semilinear PDEs on SelfSimilar Fractals
Abstract
A Laplacian may be defined on selfsimilar fractal domains in terms of a suitable selfsimilar Dirichlet form, enabling discussion of elliptic PDEs on such domains. In this context it is shown that that semilinear equations such as ∆u+u^{p}= 0, with zero Dirichlet boundary conditions, have nontrivial nonnegative solutions if 0<ν<= 2 and p>1, or if ν >2 and 1<p< (ν+ 2)/(ν 2), where ν is the ``intrinsic dimension'' or ``spectral dimension'' of the system. Thus the intrinsic dimension takes the rôle of the Euclidean dimension in the classical case in determining critical exponents of semilinear problems.
 Publication:

Communications in Mathematical Physics
 Pub Date:
 1999
 DOI:
 10.1007/s002200050703
 Bibcode:
 1999CMaPh.206..235F