Summary
In this paper, the effect of the domain geometry on the local regularity/singularity properties of the solution to the trace of a multifractional pseudodifferential equation on a fractal domain is studied. The singularity spectrum of the Gaussian solution to this type of models is trivial due to regularity assumptions on the variable order of its fractional derivatives. The theory of reproducing kernel Hilbert spaces (RKHSs) and generalized random fields is applied in this study. Specifically, the associated family of RKHSs is isomorphically identified with the trace on a compact fractal domain of a multifractional Sobolev space. The fractal defect modifies the variable order of weak-sense factional derivatives of the functions in these spaces. In the Gaussian case, random fields on fractal domains having sample paths with variable local Hölder exponent are introduced in this framework.
Keywords
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Angulo, J.M., Ruiz-Medina, M.D. (2011). Multifractional Random Systems on Fractal Domains. In: Pardo, L., Balakrishnan, N., Gil, M.Á. (eds) Modern Mathematical Tools and Techniques in Capturing Complexity. Understanding Complex Systems. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-20853-9_25
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