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Consider the following quasilinear elliptic equation $$\nabla_x (A(x,u(x),\nabla_x u(x))) + f(u,x) = 0 $$ on a bounded domain $\Omega$, augmented with homogeneous Dirichlet boundary data: $$u|_{\partial \Omega} = 0.$$

Has this problem been studied in the case of $\Omega$ being a fractal set?

In general, what papers/monographs deal with elliptic problems (even the model Laplace equation) on fractal domains?

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  • $\begingroup$ (1/2) Diffusions on fractals have been studied quite a lot. Heat kernel seems to be one of the central objects here. Some names that come to my mind: Barlow, Chen, Grigor'yan, Kumagai, Wang (I am certainly missing many more). State of the art depends on the class of fractals one is interested in: finitely ramified self-similar sets are relatively simple, Sierpiński carpets can be challenging, percolation clusters are likely somewhere in between. If you are interested, I can give some references on that. $\endgroup$
    – Mateusz Kwaśnicki
    Jan 12, 2019 at 23:28
  • $\begingroup$ (2/2) Non-linear problems are likely harder and I do not know if they have been studied. Be aware, however, that the "Laplacian" on fractals usually cannot be expressed using fractal "gradients", so the way you write the elliptic equation makes little sense on, say, the Sierpiński gasket. Unless I misunderstood your question completely. $\endgroup$
    – Mateusz Kwaśnicki
    Jan 12, 2019 at 23:30
  • $\begingroup$ @MateuszKwaśnicki Both your comments are interesting. Could you elaborate on them? $\endgroup$
    – user60665
    Jan 12, 2019 at 23:34
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    $\begingroup$ Well, this is a large, diversified area, so it is not possible to summarize all of it in a comment. And I am not an expert in this field. You may like to look at sample literature to see if this is what you need. A quick google search suggests a book Differential Equations on Fractals: A Tutorial by Strichartz (a name that definitely should have appeared in the list above). $\endgroup$
    – Mateusz Kwaśnicki
    Jan 12, 2019 at 23:55
  • $\begingroup$ @MateuszKwaśnicki Thank you. What about your second remark? That is, why the can't the fractal Laplacian be written by using fractal gradient/divergence? $\endgroup$
    – user60665
    Jan 13, 2019 at 0:23

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