Boundary Value Problems on a Half Sierpinski Gasket

Abstract

We study boundary value problems for the Laplacian on a domain \Omega consisting of the left half of the Sierpinski Gasket ( SG ), whose boundary is essentially a countable set of points X . For harmonic functions we give an explicit Poisson integral formula to recover the function from its boundary values, and characterize those that correspond to functions of finite energy. We give an explicit Dirichlet to Neumann map and show that it is invertible. We give an explicit description of the Dirichlet to Neumann spectra of the Laplacian with an exact count of the dimensions of eigenspaces. We compute the exact trace spaces on X of the L^2 and L^\infty domains of the Laplacian on SG . In terms of the these trace spaces, we characterize the functions in the L^2 and L^\infty domains of the Laplacian on \Omega that extend to the corresponding domains on SG , and give an explicit linear extension operator in terms of piecewise biharmonic functions.