Nageswari Shanmugalingam

Bio: Nageswari Shanmugalingam is an academic researcher from University of Cincinnati. The author has contributed to research in topics: Metric space & Measure (mathematics). The author has an hindex of 29, co-authored 128 publications receiving 3791 citations. Previous affiliations of Nageswari Shanmugalingam include Maynooth University & University of Texas at Austin.

Newtonian spaces: An extension of Sobolev spaces to metric measure spaces

Abstract: This paper studies a possible definition of Sobolev spaces in abstract metric spaces, and answers in the affirmative the question whether this definition yields a Banach space. The paper also explores the relationship between this definition and the Hajlasz spaces. For specialized metric spaces the Sobolev embedding theorems are proven. Different versions of capacities are also explored, and these various definitions are compared. The main tool used in this paper is the concepto of moduli of path families.

Sobolev Spaces on Metric Measure Spaces: An Approach Based on Upper Gradients

Abstract: Analysis on metric spaces emerged in the 1990s as an independent research field providing a unified treatment of first-order analysis in diverse and potentially nonsmooth settings. Based on the fundamental concept of upper gradient, the notion of a Sobolev function was formulated in the setting of metric measure spaces supporting a Poincare inequality. This coherent treatment from first principles is an ideal introduction to the subject for graduate students and a useful reference for experts. It presents the foundations of the theory of such first-order Sobolev spaces, then explores geometric implications of the critical Poincare inequality, and indicates numerous examples of spaces satisfying this axiom. A distinguishing feature of the book is its focus on vector-valued Sobolev spaces. The final chapters include proofs of several landmark theorems, including Cheeger's stability theorem for Poincare inequalities under Gromov–Hausdorff convergence, and the Keith–Zhong self-improvement theorem for Poincare inequalities.

Harmonic functions on metric spaces

Abstract: This paper explores a Dirichlet type problem on metric measure spaces The problem is to find a Sobolev-type function that minimizes the energy integral within a class of "Sobolev" functions that agree with the boundary function outside the domain of the problem This is the analogue of the Euler-Lagrange formulation in the classical Dirichlet problem It is shown that, under certain geometric constraints on the measure imposed on the metric space, such a solution exists Under the condition that the space has many rectifiable curves, the solution is unique and satisfies the weak maximum principle

Sobolev classes of Banach space-valued functions and quasiconformal mappings

Abstract: We give a definition for the class of Sobolev functions from a metric measure space into a Banach space. We give various characterizations of Sobolev classes and study the absolute continuity in measure of Sobolev mappings in the “borderline case”. We show under rather weak assumptions on the source space that quasisymmetric homeomorphisms belong to a Sobolev space of borderline degree; in particular, they are absolutely continuous. This leads to an analytic characterization of quasiconformal mappings between Ahlfors regular Loewner spaces akin to the classical Euclidean situation. As a consequence, we deduce that quasisymmetric maps respect the Cheeger differentials of Lipschitz functions on metric measure spaces with borderline Poincare inequality.

Regularity of quasi-minimizers on metric spaces

Abstract: Using the theory of Sobolev spaces on a metric measure space we are able to apply calculus of variations and define p-harmonic functions as minimizers of the p-Dirichlet integral. More generally, we study regularity properties of quasi-minimizers of p-Dirichlet integrals in a metric measure space. Applying the De Giorgi method we show that quasi-minimizers, and in particular p-harmonic functions, satisfy Harnack's inequality, the strong maximum principle, and are locally Holder continuous, if the space is doubling and supports a Poincare inequality.

Differentiability of Lipschitz Functions on Metric Measure Spaces

Abstract: Abstract. ((Without Abstract)).

Metric measure spaces with Riemannian Ricci curvature bounded from below

Abstract: In this paper, we introduce a synthetic notion of Riemannian Ricci bounds from below for metric measure spaces (X,d,m) which is stable under measured Gromov– Hausdorff convergence and rules out Finsler geometries. It can be given in terms of an enforcement of the Lott, Sturm, and Villani geodesic convexity condition for the entropy coupled with the linearity of the heat flow. Besides stability, it enjoys the same tensorization, global-to-local, and local-to-global properties. In these spaces, which we call RCD(K,∞) spaces, we prove that the heat flow (which can be equivalently characterized either as the flow associated to the Dirichlet form, or as the Wasserstein gradient flow of the entropy) satisfies Wasserstein contraction estimates and several regularity properties, in particular Bakry–Emery estimates and the L∞-Lip Feller regularization. We also prove that the distance induced by the Dirichlet form coincides with d, that the local energy measure has density given by the square of Cheeger’s relaxed slope, and, as a consequence, that the underlying Brownian motion has continuous paths. All these results are obtained independently of Poincare and doubling assumptions on the metric measure structure and therefore apply also to spaces which are not locally compact, as the infinite-dimensional ones.