Neumann conditions on fractal boundaries

We consider some elliptic boundary value problems in a self-similar ramified domain of R 2 with a fractal boundary with Laplace's equation and nonhomogeneous Neumann boundary conditions. The Hausdorff dimension of the fractal boundary is greater than one. The goal is twofold: first rigorously define the boundary value problems, second approximate the restriction of the solutions to subdomains obtained by stopping the geometric construction after a finite number of steps. For the first task, a key step is the definition of a trace operator. For the second task, a multiscale strategy based on transparent boundary conditions and on a wavelet expansion of the Neumann datum is proposed, following an idea contained in a previous work by the same authors. Error estimates are given and numerical results are presented.

[1]  Brian Street,et al.  Sampling on the Sierpinski Gasket , 2003, Exp. Math..

[2]  P. A. P. Moran,et al.  Additive functions of intervals and Hausdorff measure , 1946, Mathematical Proceedings of the Cambridge Philosophical Society.

[3]  Günter Berger,et al.  Eigenvalue Distribution of Elliptic Operators of Second Order with Neumann Boundary Conditions in a Snowflake Domain , 2000 .

[4]  Yves Achdou,et al.  A Multiscale Numerical Method for Poisson Problems in Some Ramified Domains with a Fractal Boundary , 2006, Multiscale Model. Simul..

[5]  Umberto Mosco,et al.  Energy Functionals on Certain Fractal Structures , 2003 .

[6]  Maria Rosaria Lancia,et al.  A Transmission Problem with a Fractal Interface , 2002 .

[7]  Peter W. Jones.  Quasiconformal mappings and extendability of functions in sobolev spaces , 1981 .

[8]  J. Kigami,et al.  Analysis on Fractals , 2001 .

[9]  Christophe Sabot.  Spectral properties of self-similar lattices and iteration of rational maps , 2002 .

[10]  Jaak Peetre,et al.  Function spaces on subsets of Rn , 1984 .

[11]  K. Falconer.  Techniques in fractal geometry , 1997 .

[12]  Robert S. Strichartz,et al.  The Finite Element Method on the Sierpinski Gasket , 2001 .

[13]  Yves Achdou,et al.  Diffusion and propagation problems in some ramified domains with a fractal boundary , 2006 .

[14]  B. Sapoval,et al.  Vibrations of fractal drums. , 1991, Physical review letters.

[15]  Sapoval,et al.  Vibrations of strongly irregular or fractal resonators. , 1993, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[16]  Christophe Sabot.  Electrical Networks, Symplectic Reductions, and Application to the Renormalization Map of Self-Similar Lattices , 2003 .