arXiv subjects

Nageswari Shanmugalingam

Publications and source records attributed to Nageswari Shanmugalingam.

At least 37 records · Page 2Linked to original sources

arXiv2023

Discontinuous eikonal equations in metric measure spaces

In this paper, we study the eikonal equation in metric measure spaces, where the inhomogeneous term is allowed to be discontinuous, unbounded and merely $p$-integrable in the domain with a finite $p$. For continuous eikonal equations, it is known that the notion of Monge solutions is equivalent to the standard definition of viscosity solutions. Generalizing the notion of Monge solutions in our setting, we establish uniqueness and existence results for the associated Dirichlet boundary problem. The key in our approach is to adopt a new metric, based on the optimal control interpretation, which integrates the discontinuous term and converts the eikonal equation to a standard continuous form. We also discuss the Holder continuity of the unique solution with respect to the original metric under regularity assumptions on the space and the inhomogeneous term.

math.AP↗

arXiv2023

Solving a Dirichlet problem for unbounded domains via a conformal transformation

In this paper, we solve the $p$-Dirichlet problem for Besov boundary data on unbounded uniform domains with bounded boundaries when the domain is equipped with a doubling measure satisfying a Poincaré inequality. This is accomplished by studying a class of transformations that have been recently shown to render the domain bounded while maintaining uniformity. These transformations conformally deform the metric and measure in a way that depends on the distance to the boundary of the domain and, for the measure, a parameter $p$. We show that the transformed measure is doubling and the transformed domain supports a Poincaré inequality. This allows us to transfer known results for bounded uniform domains to unbounded ones, including trace results and Adams-type inequalities, culminating in a solution to the Dirichlet problem for boundary data in a Besov class.

math.AP↗

arXiv2022

Trace and extension theorems for homogeneous Sobolev and Besov spaces for unbounded uniform domains in metric measure spaces

In this paper we fix $1\le p<\infty$ and consider $(\Om,d,μ)$ be an unbounded, locally compact, non-complete metric measure space equipped with a doubling measure $μ$ supporting a $p$-Poincaré inequality such that $\Om$ is a uniform domain in its completion $\bar\Om$. We realize the trace of functions in the Dirichlet-Sobolev space $D^{1,p}(\Om)$ on the boundary $\partial\Om$ as functions in the homogeneous Besov space $HB^α_{p,p}(\partial\Om)$ for suitable $α$; here, $\partial\Om$ is equipped with a non-atomic Borel regular measure $ν$. We show that if $ν$ satisfies a $θ$-codimensional condition with respect to $μ$ for some $0<θ<p$, then there is a bounded linear trace operator $T:D^{1,p}(\Om)\rightarrow HB^{1-θ/p}(\partial\Om)$ and a bounded linear extension operator $E:HB^{1-θ/p}(\partial\Om)\rightarrow D^{1,p}(\Om)$ that is a right-inverse of $T$.

math.FA↗

arXiv2022

Potential theory and quasisymmetric maps between compact Ahlfors regular metric measure spaces via Besov functions: preliminary

We study Besov capacities in a compact Ahlfors regular metric measure space by means of hyperbolic fillings of the space. This approach is applicable even if the space does not support any Poincaré inequalities. As an application of the Besov capacity estimates we show that if a homeomorphism between two Ahlfors regular metric measure spaces preserves, under some additional assumptions, certain Besov classes, then the homeomorphism is necessarily a quasisymmetric map.

math.MG↗

arXiv2022

Conformal transformation of uniform domains under weights that depend on distance to the boundary

The sphericalization procedure converts a Euclidean space into a compact sphere. In this note we propose a variant of this procedure for locally compact, rectifiably path-connected, non-complete, unbounded metric spaces by using conformal deformations that depend only on the distance to the boundary of the metric space. This deformation is locally bi-Lipschitz to the original domain near its boundary, but transforms the space into a bounded domain. We will show that if the original metric space is a uniform domain with respect to its completion, then the transformed space is also a uniform domain.

math.MG↗

arXiv2022

Besov class via heat semigroup on Dirichlet spaces III: BV functions and sub-Gaussian heat kernel estimates

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. Under a weak Bakry-Émery curvature type condition, which is new in this setting, this BV class is identified with a heat semigroup based Besov class. As a consequence of this identification, properties of BV functions and associated BV measures are studied in detail. In particular, we prove co-area formulas, global $L^1$ Sobolev embeddings and isoperimetric inequalities. It is shown that for nested fractals or their direct products the BV class we define is dense in $L^1$. The examples of the unbounded Vicsek set, unbounded Sierpinski gasket and unbounded Sierpinski carpet are discussed.

math.MG↗

arXiv2022

On Carrasco Piaggio's theorem characterizing quasisymmetric maps from compact doubling spaces to Ahlfors regular spaces

In this note we deconstruct and explore the components of a theorem of Carrasco Piaggio, which relates Ahlfors regular conformal gauge of a compact doubling metric space to weights on Gromov-hyperbolic fillings of the metric space. We consider a construction of hyperbolic filling that is simpler than the one considered by Carrasco Piaggio, and we determine the effect of each of the four properties postulated by Carrasco Piaggio on the induced metric on the compact metric space.

math.MG↗

arXiv2022

Neumann problems for $p$-harmonic functions, and induced nonlocal operators in metric measure spaces

Following ideas of Caffarelli and Silvestre in~\cite{CS}, and using recent progress in hyperbolic fillings, we define fractional $p$-Laplacians $(-Δ_p)^θ$ with $0<θ<1$ on any compact, doubling metric measure space $(Z,d,ν)$, and prove existence, regularity and stability for the non-homogenous non-local equation $(-Δ_p)^θu =f.$ These results, in turn, rest on the new existence, global Hölder regularity and stability theorems that we prove for the Neumann problem for $p$-Laplacians $Δ_p$, $1<p<\infty$, in bounded domains of measure metric spaces endowed with a doubling measure that supports a Poincaré inequality. Our work also extends beyond the compact setting, and includes as special cases much of the previous results by other authors in the Euclidean, Riemannian and Carnot group settings. Unlike other recent contributions in the metric measure spaces context, our work does not rely on the assumption that $(Z,d,ν)$ supports a Poincaré inequality.

math.AP↗

arXiv2021

Classification of metric measure spaces and their ends using $p$-harmonic functions

By seeing whether a Liouville type theorem holds for positive, bounded, and/or finite energy $p$-harmonic and $p$-quasiharmonic functions, we classify proper metric spaces equipped with a locally doubling measure supporting a local $p$-Poincaré inequality. Similar classifications have earlier been obtained for Riemann surfaces and Riemannian manifolds. We also study the inclusions between these classes of metric measure spaces, and their relationship to the $p$-hyperbolicity of the metric space and its ends. In particular, we characterize spaces that carry nonconstant $p$-harmonic functions with finite energy as spaces having at least two well-separated $p$-hyperbolic sequences. We also show that every such space $X$ has a function $f \notin L^p(X) + \mathbb{R} $ with finite $p$-energy.

math.MG↗

arXiv2021

Extension and trace results for doubling metric measure spaces and their hyperbolic fillings

In this paper we study connections between Besov spaces of functions on a compact metric space $Z$, equipped with a doubling measure, and the Newton--Sobolev space of functions on a uniform domain $X_\varepsilon$. This uniform domain is obtained as a uniformization of a (Gromov) hyperbolic filling of $Z$. To do so, we construct a family of hyperbolic fillings in the style of the work of Bonk and Kleiner and the work of Bourdon and Pajot. Then for each parameter $β>0$ we construct a lift $μ_β$ of the doubling measure $ν$ on $Z$ to $X_\varepsilon$, and show that $μ_β$ is doubling and supports a $1$-Poincaré inequality. We then show that for each $θ$ with $0<θ<1$ and $p\ge 1$ there is a choice of $β=p(1-θ)\logα$ such that the Besov space $B^θ_{p,p}(Z)$ is the trace space of the Newton--Sobolev space $N^{1,p}(X_\varepsilon,μ_β)$ when $\varepsilon=\logα$. Finally, we exploit the tools of potential theory on $X_\varepsilon$ to obtain fine properties of functions in $B^θ_{p,p}(Z)$, such as their quasicontinuity and quasieverywhere existence of $L^q$-Lebesgue points with $q=s_νp/(s_ν-pθ)$, where $s_ν$ is a doubling dimension associated with the measure $ν$ on $Z$. Applying this to compact subsets of Euclidean spaces improves upon a result of Netrusov in $\mathbb{R}^n$.

math.MG↗

arXiv2020

Regularity of Solutions to the Fractional Cheeger-Laplacian on Domains in Metric Spaces of Bounded Geometry

We study existence, uniqueness, and regularity properties of the Dirichlet problem related to fractional Dirichlet energy minimizers in a complete doubling metric measure space $(X,d_X,μ_X)$ satisfying a $2$-Poincaré inequality. Given a bounded domain $Ω\subset X$ with $μ_X(X\setminusΩ)>0$, and a function $f$ in the Besov class $B^θ_{2,2}(X)\cap L^2(X)$, we study the problem of finding a function $u\in B^θ_{2,2}(X)$ such that $u=f$ in $X\setminusΩ$ and $\mathcal{E}_θ(u,u)\le \mathcal{E}_θ(h,h)$ whenever $h\in B^θ_{2,2}(X)$ with $h=f$ in $X\setminusΩ$. We show that such a solution always exists and that this solution is unique. We also show that the solution is locally Hölder continuous on $Ω$, and satisfies a non-local maximum and strong maximum principle. Part of the results in this paper extend the work of Caffarelli and Silvestre in the Euclidean setting and Franchi and Ferrari in Carnot groups.

math.AP↗

arXiv2020

Besov class via heat semigroup on Dirichlet spaces II: BV functions and Gaussian heat kernel estimates

We introduce the class of bounded variation (BV) functions in a general framework of strictly local Dirichlet spaces with doubling measure. Under the 2-Poincaré inequality and a weak Bakry-Émery curvature type condition, this BV class is identified with the heat semigroup based Besov class $\mathbf{B}^{1,1/2}(X)$ that was introduced in our previous paper. Assuming furthermore a quasi Bakry-Émery curvature type condition, we identify the Sobolev class $W^{1,p}(X)$ with $\mathbf{B}^{p,1/2}(X)$ for $p>1$. Consequences of those identifications in terms of isoperimetric and Sobolev inequalities with sharp exponents are given.

math.MG↗

arXiv2020

Traces and extensions of certain weighted Sobolev spaces on $\mathbb{R}^n$ and Besov functions on Ahlfors regular compact subsets of $\mathbb{R}^n$

The focus of this paper is on Ahlfors $Q$-regular compact sets $E\subset\mathbb{R}^n$ such that, for each $Q-2<α\le 0$, the weighted measure $μ_α$ given by integrating the density $ω(x)=\text{dist}(x, E)^α$ yields a Muckenhoupt $\mathcal{A}_p$-weight in a ball $B$ containing $E$. For such sets $E$ we show the existence of a bounded linear trace operator acting from $W^{1,p}(B,μ_α)$ to $B^θ_{p,p}(E, \mathcal{H}^Q\vert_E)$ when $0<θ<1-\tfrac{α+n-Q}{p}$, and the existence of a bounded linear extension operator from $B^θ_{p,p}(E, \mathcal{H}^Q\vert_E)$ to $W^{1,p}(B, μ_α)$ when $1-\tfrac{α+n-Q}{p}\le θ<1$. We illustrate these results with $E$ as the Sierpiński carpet, the Sierpiński gasket, and the von Koch snowflake.

math.CV↗

arXiv2020

Equivalence of solutions of eikonal equation in metric spaces

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 rectifiably connected metric spaces. The notions considered are that of curve-based viscosity solutions, slope-based viscosity solutions, and Monge solutions. By using the induced intrinsic (path) metric, we reduce the metric space to a length space and show the equivalence of these solutions to the associated Dirichlet boundary problem. Without utilizing the boundary data, we also localize our argument and directly prove the equivalence for the definitions of solutions. Regularity of solutions related to the Euclidean semi-concavity is discussed as well.

math.AP↗

arXiv2020

Rough isometry between Gromov hyperbolic spaces and uniformization

In this note we show that given two complete geodesic Gromov hyperbolic spaces that are roughly isometric and $\varepsilon>0$, either the uniformization of both spaces with parameter $\varepsilon$ results in uniform domains, or else neither uniformized space is a uniform domain. The terminology of "uniformization" is from the work of Bonk, Heinonen and Koskela, where it is shown that the uniformization, with parameter $\varepsilon>0$, of a complete geodesic Gromov hyperbolic space results in a uniform domain provided $\varepsilon$ is small enough.

math.MG↗

arXiv2020

Besov class via heat semigroup on Dirichlet spaces I: Sobolev type inequalities

We introduce heat semigroup-based Besov classes in the general framework of Dirichlet spaces. General properties of those classes are studied and quantitative regularization estimates for the heat semigroup in this scale of spaces are obtained. As a highlight of the paper, we obtain a far reaching $L^p$-analogue, $p \ge 1$, of the Sobolev inequality that was proved for $p=2$ by N. Varopoulos under the assumption of ultracontractivity for the heat semigroup. The case $p=1$ is of special interest since it yields isoperimetric type inequalities.

math.FA↗

arXiv2020

Asymptotic behavior of BV functions and sets of finite perimeter in metric measure spaces

In this paper, we study the asymptotic behavior of BV functions in complete metric measure spaces equipped with a doubling measure supporting a $1$-Poincaré inequality. We show that at almost every point $x$ outside the Cantor and jump parts of a BV function, the asymptotic limit of the function is a Lipschitz continuous function of least gradient on a tangent space to the metric space based at $x$. We also show that, at co-dimension $1$ Hausdorff measure almost every measure-theoretic boundary point of a set $E$ of finite perimeter, there is an asymptotic limit set $(E)_\infty$ corresponding to the asymptotic expansion of $E$ and that every such asymptotic limit $(E)_\infty$ is a quasiminimal set of finite perimeter. We also show that the perimeter measure of $(E)_\infty$ is Ahlfors co-dimension $1$ regular.

math.MG↗

arXiv2019

Analysis on Metric Spaces

This note is a survey of Analysis on Metric spaces, in connection with the upcoming AMS Mathematics Research Communities program in June 2020.

math.CV↗