46 citations · 339 across the 63 of their papers we have counts for
18 papers · 1 filter
Approximation of harmonic functions on metric measure spaces of controlled geometry via discrete graphs
Almaz Butaev, Liangbing Luo, Nageswari Shanmugalingam
Given a complete doubling metric measure space that supports a -Poincaré inequality, we approximate harmonic functions on a bounded domain with a prescribed Newton-Sobol…
Solving Dirichlet problem on unbounded uniform domains by using sphericalization techniques
Riikka Korte, Sari Rogovin, Nageswari Shanmugalingam +1
Within the setting of metric spaces equipped with a doubling measure and supporting a -Poincaré inequality, establishing existence of solutions to Dirichlet problem in a bounded…
Sharp Hölder regularity of weak solutions of the Neumann problem and applications to nonlocal PDE in metric measure spaces
Luca Capogna, Ryan Gibara, Riikka Korte +1
We prove global Hölder regularity result for weak solutions to a PDE of -Laplacian type with a measure as non-homogeneous term: \[ -\text{div}\!\left( |\nab…
Fractional -Laplacians via Neumann problems in unbounded metric measure spaces
Luca Capogna, Ryan Gibara, Riikka Korte +1
We prove well-posedness, Harnack inequality and sharp regularity of solutions to a fractional -Laplace non-homogeneous equation , with , , for d…
Well-posedness of Dirichlet boundary value problems for reflected fractional -Laplace-type inhomogeneous equations in compact doubling metric measure spaces
Josh Kline, Feng Li, Nageswari Shanmugalingam
In this paper we consider the setting of a locally compact, non-complete metric measure space equipped with a doubling measure , under the condition that the boundary…
On the Dirichlet-to-Neumann Map for the -Laplacian on a Metric Measure Space
Ryan Gibara, Nageswari Shanmugalingam
In this note, we construct a Dirichlet-to-Neumann map, from a Besov space of functions, to the dual of this class. The Besov spaces are of functions on the boundary of a bounded, l…
Discontinuous eikonal equations in metric measure spaces
Qing Liu, Nageswari Shanmugalingam, Xiaodan Zhou
In this paper, we study the eikonal equation in metric measure spaces, where the inhomogeneous term is allowed to be discontinuous, unbounded and merely -integrable in the domai…
Solving a Dirichlet problem for unbounded domains via a conformal transformation
Ryan Gibara, Riikka Korte, Nageswari Shanmugalingam
In this paper, we solve the -Dirichlet problem for Besov boundary data on unbounded uniform domains with bounded boundaries when the domain is equipped with a doubling measure s…
Neumann problems for -harmonic functions, and induced nonlocal operators in metric measure spaces
Luca Capogna, Josh Kline, Riikka Korte +2
Following ideas of Caffarelli and Silvestre in~\cite{CS}, and using recent progress in hyperbolic fillings, we define fractional -Laplacians with on any compa…
Regularity of Solutions to the Fractional Cheeger-Laplacian on Domains in Metric Spaces of Bounded Geometry
Sylvester Eriksson-Bique, Gianmarco Giovannardi, Riikka Korte +2
We study existence, uniqueness, and regularity properties of the Dirichlet problem related to fractional Dirichlet energy minimizers in a complete doubling metric measure space $(X…
Equivalence of solutions of eikonal equation in metric spaces
Qing Liu, Nageswari Shanmugalingam, Xiaodan Zhou
In this paper we prove the equivalence between some known notions of solutions to the eikonal equation and more general analogs of the Hamilton-Jacobi equations in complete and rec…
Domains in metric measure spaces with boundary of positive mean curvature, and the Dirichlet problem for functions of least gradient
Panu Lahti, Lukas Maly, Nageswari Shanmugalingam +1
We study the geometry of domains in complete metric measure spaces equipped with a doubling measure supporting a -Poincaré inequality. We propose a notion of \emph{domain with b…
Dichotomy of global capacity density in metric measure spaces
Hiroaki Aikawa, Anders Björn, Jana Björn +1
The variational capacity cap_p in Euclidean spaces is known to enjoy the density dichotomy at large scales, namely that for every subset E of R^n, inf_{x in R^n} (cap_p(E \cap B(x,…
Notions of Dirichlet problem for functions of least gradient in metric measure spaces
Riikka Korte, Panu Lahti, Xining Li +1
We study two notions of Dirichlet problem associated with BV energy minimizers (also called functions of least gradient) in bounded domains in metric measure spaces whose measure i…
Fine properties and a notion of quasicontinuity for BV functions on metric spaces
Panu Lahti, Nageswari Shanmugalingam
On a metric space equipped with a doubling measure supporting a Poincaré inequality, we show that given a BV function, discarding a set of small -capacity makes the function con…
Characterizations of sets of finite perimeter using heat kernels in metric spaces
Niko Marola, Michele Miranda, Nageswari Shanmugalingam
The overarching goal of this paper is to link the notion of sets of finite perimeter (a concept associated with -spaces) and the theory of heat semigroups (a concept relat…
Boundary measures, generalized Gauss-Green formulas, and mean value property in metric measure spaces
Niko Marola, Michele Miranda, Nageswari Shanmugalingam
We study mean value properties of harmonic functions in metric measure spaces. The metric measure spaces we consider have a doubling measure and support a (1,1)- Poincaré inequalit…
Regularity of sets with quasiminimal boundary surfaces in metric spaces
Juha Kinnunen, Riikka Korte, Andrew Lorent +1
This paper studies regularity of perimiter quasiminimizing sets in metric measure spaces with a doubling measure and a Poincare inequality. The main result shows that the measure t…