Papers by Nageswari Shanmugalingam
In this paper we show existence of traces of functions of bounded variation on the boundary of a ... more In this paper we show existence of traces of functions of bounded variation on the boundary of a certain class of domains in metric measure spaces equipped with a doubling measure supporting a 1-Poincaré inequality, and obtain L 1 estimates of the trace functions. In contrast with the treatment of traces given in other papers on this subject, the traces we consider do not require knowledge of the function in the exterior of the domain. We also establish a Maz'yatype inequality for functions of bounded variation that vanish on a set of positive capacity.
In this note we study the Dirichlet problem associated with a version of prime end boundary of a ... more In this note we study the Dirichlet problem associated with a version of prime end boundary of a bounded domain in a complete metric measure space equipped with a doubling measure supporting a Poincare inequality. We show the resolutivity of functions that are continuous on the prime end boundary and are Lipschitz regular when restricted to the subset of all prime ends whose impressions are singleton sets. We also consider a new notion of capacity adapted to the prime end boundary, and show that bounded perturbations of such functions on subsets of the prime end boundary with zero capacity are resolutive and that their Perron solutions coincide with the Perron solution of the original functions. We also describe some examples which demonstrate the efficacy of the prime end boundary approach in obtaining new results even for the classical Dirichlet problem for some Euclidean domains.
On relatively compact domains in metric measure spaces we construct singular functions that play ... more On relatively compact domains in metric measure spaces we construct singular functions that play the role of Green functions of the p-Laplacian. We give a characterization of metric spaces that support a global version of such singular function, in terms of capacity estimates at infinity of such metric spaces. In addition, when the measure of the space is locally Q-regular, we study quasiconformal invariance property associated with the existence of global singular functions.
Potential Analysis, 2014
ABSTRACT In this note we study the Dirichlet problem associated with a version of prime end bound... more ABSTRACT In this note we study the Dirichlet problem associated with a version of prime end boundary of a bounded domain in a complete metric measure space equipped with a doubling measure supporting a Poincare inequality. We show the resolutivity of functions that are continuous on the prime end boundary and are Lipschitz regular when restricted to the subset of all prime ends whose impressions are singleton sets. We also consider a new notion of capacity adapted to the prime end boundary, and show that bounded perturbations of such functions on subsets of the prime end boundary with zero capacity are resolutive and that their Perron solutions coincide with the Perron solution of the original functions. We also describe some examples which demonstrate the efficacy of the prime end boundary approach in obtaining new results even for the classical Dirichlet problem for some Euclidean domains.
Archive for Rational Mechanics and Analysis, 2014
ABSTRACT The aim of this paper is two-fold: First, we obtain a better understanding of the intrin... more ABSTRACT The aim of this paper is two-fold: First, we obtain a better understanding of the intrinsic distance of diffusion processes. Precisely, (i) for all $n\ge1$, the diffusion matrix $A$ is weak upper semicontinuous on $\Omega$ if and only if the intrinsic differential and the local intrinsic distance structures coincide; (ii) if $n=1$, or if $n\ge2$ and $A$ is weak upper semicontinuous on $\Omega$, the intrinsic distance and differential structures always coincide; (iii) if $n\ge2$ and $A$ fails to be weak upper semicontinuous on $\Omega$, the (non-) coincidence of the intrinsic distance and differential structures depend on the geometry of the non-weak-upper-semicontinuity set of $A$. Second, for an arbitrary diffusion matrix $A$, we show that the intrinsic distance completely determines the absolute minimizer of the corresponding $L^\infty$-variational problem, and then obtain the existence and uniqueness for given boundary data. We also give an example of a diffusion matrix $A$ for which there is an absolute minimizer that is not of class $C^1$. When $A$ is continuous, we also obtain the linear approximation property of the absolute minimizer.
Calculus of Variations and Partial Differential Equations, 2015
ABSTRACT On metric spaces supporting a geometric version of a Semmes family of curves, we provide... more ABSTRACT On metric spaces supporting a geometric version of a Semmes family of curves, we provide a Reshetnyak-type characterization of functions of bounded variation in terms of the total variation on such a family of curves. We then use this characterization to obtain a Federer-type characterization of sets of finite perimeter, that is, we show that a measurable set is of finite perimeter if and only if the Hausdorff measure of its measure theoretic boundary is finite. We present a construction of a geometric Semmes family of curves in the first Heisenberg group.
Journal of Functional Analysis, 2014
International Mathematical Series, 2009
We investigate geometric properties of those planar domains that are extension for functions with... more We investigate geometric properties of those planar domains that are extension for functions with bounded variation.We start from a characterization of such domains given by Burago–Maz'ya and prove that a bounded, simply connected domain is a BV -extension domain if and only if its com- plement is quasiconvex. We further prove that the extension property is a bi-Lipschitz invariant and
Revista Matemática Iberoamericana, 2000
We prove that a locally compact metric space that supports a doubling measure and a weak p-Poinca... more We prove that a locally compact metric space that supports a doubling measure and a weak p-Poincaré inequality for some 1 ≤ p < ∞ is a MEC p -space. The methods developed for this purpose include measurability considerations and lead to interesting consequences. For example, we verify that each extended real valued function having a p-integrable upper gradient is locally p-integrable.
Revista Matemática Complutense, 2014
We study pointwise properties of functions of bounded variation on a metric space equipped with a... more We study pointwise properties of functions of bounded variation on a metric space equipped with a doubling measure and supporting a Poincaré inequality. In particular, we obtain a Lebesgue type result for BV functions. We also study approximations by Lipschitz continuous functions and a version of the Leibniz rule. We give examples which show that our results are optimal for BV functions in this generality.
Potential Analysis, 2014
ABSTRACT We study the quasiadditivity property (a version of superadditivity with a multiplicativ... more ABSTRACT We study the quasiadditivity property (a version of superadditivity with a multiplicative constant) of variational capacity in metric spaces with respect to Whitney type covers. We characterize this property in terms of a Mazya type capacity condition, and also explore the close relation between quasiadditivity and Hardy&#39;s inequality.
Mathematische Zeitschrift, 2010
We present an easy proof that p-Hardy's inequality implies uniform p-fatness of the boundary when... more We present an easy proof that p-Hardy's inequality implies uniform p-fatness of the boundary when p = n. The proof works also in metric space setting and demonstrates the self-improving phenomenon of the p-fatness. We also explore the relationship between p-fatness, p-Hardy inequality, and the uniform perfectness for all p ≥ 1, and demonstrate that in the Ahlfors Q-regular metric measure space setting with p = Q, these three properties are equivalent. When p = 2, our results are new even in the Euclidean setting.
Mathematische Nachrichten, 2010
Let X = (X, d, µ) be a doubling metric measure space. For 0 < α < 1, 1 ≤ p, q < ∞, we define semi... more Let X = (X, d, µ) be a doubling metric measure space. For 0 < α < 1, 1 ≤ p, q < ∞, we define semi-norms
Mathematical Research Letters, 2012
ABSTRACT We study the L ∞ -variational problem associated to a general regular, strongly local Di... more ABSTRACT We study the L ∞ -variational problem associated to a general regular, strongly local Dirichlet form.We show that the intrinsic distance determines the absolute minimizer (infinite harmonic function) of the corresponding L ∞ -functional. This leads to the existence and uniqueness of the absolute minimizer on a bounded domain, given a continuous boundary data. Applying this, we also obtain that an infinity harmonic function on ℝ n may be the minimizer for several different variational problems. Finally, we apply our results to Carnot-Carathéodory spaces.
Journal of Functional Analysis, 2003
We establish the Lipschitz continuity of Cheeger-harmonic functions in certain metric measure spa... more We establish the Lipschitz continuity of Cheeger-harmonic functions in certain metric measure spaces. Examples are given to illustrate the necessity of our assumptions on these spaces.
Journal de Mathématiques Pures et Appliquées, 2010
We study the existence of a set with minimal perimeter that separates two disjoint sets in a metr... more We study the existence of a set with minimal perimeter that separates two disjoint sets in a metric measure space equipped with a doubling measure and supporting a Poincaré inequality. A measure constructed by De Giorgi is used to state a relaxed problem, whose solution coincides with the solution to the original problem for measure theoretically thick sets. Moreover, we study properties of the De Giorgi measure on metric measure spaces and show that it is comparable to the Hausdorff measure of codimension one. We also explore the relationship between the De Giorgi measure and the variational capacity of order one. The theory of functions of bounded variation on metric spaces is used extensively in the arguments.
Indiana University Mathematics Journal, 2000
Indiana University Mathematics Journal, 2008
The purpose of this work is to study regularity of Sobolev functions on metric measure spaces equ... more The purpose of this work is to study regularity of Sobolev functions on metric measure spaces equipped with a doubling measure and supporting a weak Poincaré inequality. We show that every Sobolev function whose gradient is integrable to power one has Lebesgue points outside a set of 1-capacity zero. We also show that 1-capacity is equivalent to the Hausdorff content of codimension one and study characterizations of 1-capacity in terms of Frostman's lemma and functions of bounded variation. As the main technical tool, we prove a metric space version of Gustin's boxing inequality. Our proofs are based on covering arguments and functions of bounded variation. Perimeter measures, isoperimetric inequalities and coarea formula play an essential role in our approach.
Calculus of Variations and Partial Differential Equations, 2012
We give an intrinsic characterization of the property that the zero-extension of a Newtonian func... more We give an intrinsic characterization of the property that the zero-extension of a Newtonian function, defined on an open set in a doubling metric measure space supporting a strong relative isoperimetric inequality, belongs to the Newtonian space on the entire metric space. The theory of functions of bounded variation is used extensively in the argument and we also provide a structure theorem for sets of finite perimeter under the assumption of a strong relative isoperimetric inequality. The characterization is used to prove a strong version of quasicontinuity of Newtonian functions.
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Papers by Nageswari Shanmugalingam