Seminar in Geometric Function and Mapping Theory

Iván Caamaño Aldemunde (IM PAN)

TBA

Abstract: TBA

Sławomir Kolasiński (MIM UW)

Quadratic Flatness and Regularity for Codimension-One Varifolds with Bounded Anisotropic Mean Curvature

Abstract: Let Ω be an open set in a Euclidean space X of dimension (n+1) and ϕ be a uniformly convex smooth norm on X. Consider an n-dimensional unit-density varifold V in Ω, whose generalised mean curvature vector, computed with respect to ϕ, is bounded. Assume also that the n-dimensional Hausdorff measure restricted to the support Σ of V is absolutely continuous with respect to the weight measure of V. In my recent work with Mario Santilli (arXiv:2507.18357), we showed that there exists an open and dense subset R of Σ at points of which one can touch Σ by two mutually tangent balls. At points of R we get quadratic height decay and we then apply Allard's 1986 regularity theorem to show that these points are actually regular points of class (1,α) for any 0 < α < 1. We prove also that R is almost equal to a subset Σ* of points of Σ, where at least one blow-up limit (in the sense of Painlevé-Kuratowski) is not the whole space X. The condition on the blow-up limit seems very weak and does not entail any regularity a priori; hence, there is hope that in many cases Σ* almost equals Σ. In my talk I shall outline the history of the problem of regularity for varifolds satisfying bounds on anisotropic first variation and I shall present main ingredients of the proof of the above result.

Marcin Walicki (MiNI PW) 

The Monotonicity formula in the Heisenberg group

Abstract: We explore the Bochner identity and its implications in the setting of the Heisenberg group. We introduce a functional defined in terms of the horizontal gradient associated with the right-invariant vector fields and establish a monotonicity formula for harmonic functions.  Interestingly, we show that an analogous result does not hold when the functional is defined using the more natural choice of the horizontal gradient, the one generated by the left-invariant vector fields.

The talk is based on „A note on monotonicity and Bochner formulas in Carnot groups” by Nicola Garofalo.

Marcin Gryszówka (MIM UW) 

Elliptic operators in rough sets

Abstract:  In the recent years there were a lot of developments in the research on the solutions of elliptic equations in domains with regularity worse than Lipschitz. I will talk about a recent paper Elliptic operators in rough sets and the Dirichlet problem with boundary data in Holder spaces by Cao, Hidalgo-Palencia, Martell, Prisuelos-Arribas, Zhao. The authors prove well-posedness of the Dirichlet problem for sets satisfying the capacity density condition which are bounded or unbounded with unbounded boundary. For unbounded sets with bounded boundary they prove existence of the solution and its non-uniqueness. I will present the conditions used in this paper and ideas of the proof of the result.