Traditionally, theories of “Sobolev” spaces on metric spaces have used local Lipschitz constants as a substitute for the gradient of functions. However, a recent study by Kajino and Murugan revealed that such an idea does not work for a class of self-similar sets including the planar Sierpinski carpet. The notion of conductive homogeneity was proposed to construct a counterpart of Sobolev spaces and Sobolev p-energy even for such cases. In this talk, I will review the method of construction of Sobolev spaces under the conductive homogeneity and give a class of regular polygon-based self-similar sets having the conductive homogeneity. Our condition is the local symmetry of the space with some (or no) global symmetry. In particular, we show that any locally symmetric triangle-based self-similar sets possess the conductive homogeneity. This is joint work with Y. Ota.[-]

This talk is focused on the Radon-Carleman Problem, dealing with computing and/or estimating the essential norm and/or the Fredholm radius of singular integral operators of double layer type associated with elliptic partial dfferential operators, on function spaces naturally intervening in the formulation of boundary value problems for the said operator in a given domain. The main goal is to monitor how the geometry of the domain affects the complexity of this type of study and to present a series of results in increasingly more irregular settings, culminating with that of uniformly rectifiable domains.
This is based on joint work with Dorina Mitrea and Marius Mitrea from Baylor University, which has recently appeared in volume V of our Geometric Harmonic Analysis research monograph series in Developments in Mathematics, Springer.[-]

An elegant theorem by J.L. Lions establishes well-posedness of non-autonomous evolutionary problems in Hilbert spaces which are defined by a non-autonomous form. However a regularity problem remained open for many years. We give a survey on positive and negative (partially very recent) results. One of the positive results can be applied to an evolutionary network which has been studied by Dominik Dier and Marjeta Kramar jointly with the speaker. It is governed by non-autonomous Kirchhoff conditions at the vertices of the graph. Also the diffusion coefficients may depend on time. Besides existence and uniqueness long-time behaviour can be described. When conductivity and diffusion coefficients match (so that mass is conserved) the solutions converge to an equilibrium.[-]

Joint work with Stefano Decio, Max Engelstein, Mario Michetti, and Svitlana Mayboroda. The Robin boundary condition is $\frac{1}{a} \frac{\partial u}{\partial n}+u=f$ on the boundary of a domain $U$, and we claim that for $0< a< +\infty$, the corresponding harmonic measure is mutually absolutely continuous with respect to surface measure. Here (we hope we will have finished checking that) we can consider any bounded domain $U$ in $\mathbb{R}^n$ whose boundary is Ahlfors regular of dimension $d$, $n-2< d< n$, with nontangential access. The Robin condition is then to be taken weakly, and surface measure becomes Hausdorff measure.[-]

We consider the numerical evaluation of integrals with respect to self-similar measures supported on fractal sets, with a weakly singular integrand of loga-rithmic or algebraic type. We show that, in many cases, the self-similarity of the measures, combined with the homogeneity properties of the integrand, can be exploited to express the singular integral exactly in terms of regular inte-grals, which can be readily approximated numerically using e.g. a composite barycentre rule. Our approach applies to measures supported on many well-known fractals including Cantor sets and dusts, the Sierpinski triangle, carpet and tetrahedron, the Vicsek fractal, and the Koch snowflake. We illustrate our approach via numerical examples computed using our IFSIntegrals.jl Julia code. This is joint work with Andrew Gibbs, Botond Major and Andrea Moiola.[-]