Abstract
We consider linear partial differential equations on resistance spaces that are uniformly elliptic and parabolic in the sense of quadratic forms and involve abstract gradient and divergence terms. Our main interest is to provide graph and metric graph approximations for their unique solutions. For families of equations with different coefficients on a single compact resistance space we prove that solutions have accumulation points with respect to the uniform convergence in space, provided that the coefficients remain bounded. If in a sequence of equations the coefficients converge suitably, the solutions converge uniformly along a subsequence. For the special case of local resistance forms on finitely ramified sets we also consider sequences of resistance spaces approximating the finitely ramified set from within. Under suitable assumptions on the coefficients (extensions of) linearizations of the solutions of equations on the approximating spaces accumulate or even converge uniformly along a subsequence to the solution of the target equation on the finitely ramified set. The results cover discrete and metric graph approximations, and both are discussed.
Highlights
This allows to establish a Dirichlet form based analysis [15,27,78], with respect to a given volume measure, and in particular, studies of partial differential equations on fractals [8,64,92]. These results and many later developments based on them are motivated by a considerable body of modern research in physics suggesting that in specific situations fractal models may be much more adequate than classical ones. The difficulty in this type of analysis comes from the fact that on fractals many tools from traditional calculus are not available
The discussion of first order terms is of rather abstract nature, because on most fractals there is no obvious candidate for a gradient operator; instead, it has to be constructed from a given bilinear form in a subsequent step [17,18,50,55]. (An intuitive argument why this construction cannot be trivial is the fact that for self-similar fractals, endowed with natural Hausdorff type volume measures, volume and energy are typically singular [14,40,41,43].) For a study of, say, counterparts of second order equations [31, Section 8], involving abstract gradient and divergence terms, it seems desirable to establish results which indicate that the equations have the correct physical meaning
For certain local resistance forms on finitely ramified sets, [55,96], we introduce an approximation scheme along varying spaces, general enough to accommodate both discrete and metric graph approximations
Summary
For several classes of fractal spaces, such as p.c.f. self-similar sets [12,61,62,63,64,74,75], classical Sierpinski carpets [9,11], certain Julia sets [89], Laaksø spaces [90], diamond lattice type fractals [1,3,35], and certain random fractals [33,34], the existence of resistance forms in the sense of [65,67] has been proved. The construction of resistance forms itself is based on discrete approximations [61,62,63,64], and in symmetric respectively self-adjoint situations this can be used to obtain approximation results on the level of resistance forms [19], or Dirichlet forms [86,87] In the latter case the dynamics of a partial differential equation of elliptic or parabolic type for self-adjoint operators comes into play, and it can be captured using spectral convergence results [80,88], possibly along varying Hilbert spaces [76,85]. There the approach from [5] is used, which
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
