by Malcolm Gabbard, Carlos Lima, Gamal Mograby, Luke G. Rogers, Alexander Teplyaev.

We study the discretization of a Dirichlet form on the Koch snowflake domain and its boundary with the property that both the interior and the boundary can support positive energy. Continue reading →

by Caitlin M. Davis, Laura A. LeGare, Cory W. McCartan, Luke G. Rogers

We study the analogue of a convex interpolant of two sets on Sierpinski gaskets and an associated notion of measure transport. Continue reading →

by Patricia Alonso Ruiz, Fabrice Baudoin, Li Chen, Luke Rogers, Nageswari Shanmugalingam, Alexander Teplyaev

We study $$L^p$$ Besov critical exponents and isoperimetric and Sobolev inequalities associated with fractional Laplacians on metric measure spaces. Continue reading →

by Luke Brown, Giovanni Ferrer, Gamal Mograby, Luke G. Rogers, Karuna Sangam

We consider criteria for the differentiability of functions with continuous Laplacian on the Sierpinski Gasket and its higher-dimensional variants $$SG_N$$, N>3, Continue reading →

by Antoni Brzoska, Courtney George, Samantha Jarvis, Luke G. Rogers, Alexander Teplyaev

We provide the foundation of the spectral analysis of the Laplacian on the orbital Schreier graphs of the basilica group, the iterated monodromy group of the quadratic polynomial $$z^2−1$$. Continue reading →

by Patricia Alonso-Ruiz, Fabrice Baudoin, Li Chen, Luke Rogers, Nageswari Shanmugalingam, Alexander Teplyaev

With a view toward fractal spaces, by using a Korevaar-Schoen space approach, we introduce the class of bounded variation (BV) functions in a general framework of strongly local Dirichlet spaces with a heat kernel satisfying sub-Gaussian estimates. Continue reading →

by Patricia Alonso-Ruiz, Fabrice Baudoin, Li Chen, Luke Rogers, Nageswari Shanmugalingam, Alexander Teplyaev

We develop a theory of bounded variation functions and Besov spaces in abstract Dirichlet spaces which unifies several known examples and applies to new situations, including fractals. Continue reading →

by Antoni Brzoska, Aubrey Coffey, Madeline Hansalik, Stephen Loew, Luke G. Rogers

We adapt the well-known spectral decimation technique for computing spectra of Laplacians on certain symmetric self-similar sets to the case of magnetic Schrödinger operators and work through this method completely for the diamond lattice fractal. Continue reading →