Adimurthi

Bio: Adimurthi is an academic researcher from Tata Institute of Fundamental Research. The author has contributed to research in topics: Sobolev space & Bounded function. The author has an hindex of 28, co-authored 67 publications receiving 2573 citations. Previous affiliations of Adimurthi include Indian Institute of Science.

An improved Hardy-Sobolev inequality and its application

Abstract: For\Omega \subset $IR^n$,n\geq 2, a bounded domain, and for 1 < p < n, we improve the Hardy-Sobolev inequality, by adding a term with a singular weight of the type \frac{1}{log(1/|x|)}$^2$ . We show that this weight function is optimal in the sense that the inequality fails for any other weight function more singular than this one. Moreover, we show that a series of finite terms can be added to improve the Hardy-Sobolev inequality, which answers a question of Brezis-Vazquez. Finally, we use this result to analyze the behaviour of the first eigenvalue of the operator L\mu\omega := -(div(| abla\upsilon|{p-2} abla\upilson)as \mu increases to \frac{n-p}{p}$^p$ for 1 < p < n.

A singular Moser-Trudinger embedding and its applications

Abstract: Let Ω be a bounded domain in \({\mathbb{R}}^{n}\) , we prove the singular Moser-Trudinger embedding: \(\mathop {\sup\limits_{\parallel u\parallel \leqslant 1\Omega } \int {\frac{{e^{\alpha |u|^{\frac{n} {{n - 1}}} } }}{{|x|^\beta }}} } 0,\beta \in [0,n),u \in W_0^{1,n} (\Omega )\) and \(\parallel u\parallel = \left({\int\limits_\Omega {| abla u|^n } } \right)^\frac{1}{n}\) . We will also study the corresponding critical exponent problem.

Blow-up Analysis in Dimension 2 and a Sharp Form of Trudinger–Moser Inequality

Abstract: This paper deals with an improvement of the Trudinger–Moser inequality associated to the embedding of the standard Sobolev space into Orlicz spaces for Ω a smooth bounded domain in ℝ2. The inequality proved here gives in particular precise informations on a previous result obtained by Lions and can be very useful in the study of lack of compactness of the embedding of into {exp(4πu 2) ∈ L 1(Ω)}. We also provide a general asymptotic analysis for sequences of solutions to elliptic PDE's with critical Sobolev growth which blow up at some point. We obtain in particular a result which is well-known in higher dimensions: the concentration points are located at critical points of the regular part of the Green function of the linear operator involved in the equation.

Optimal entropy solutions for conservation laws with discontinuous flux-functions

Abstract: We deal with a single conservation law in one space dimension whose flux function is discontinuous in the space variable and we introduce a proper framework of entropy solutions. We consider a large class of fluxes, namely, fluxes of the convex-convex type and of the concave-convex (mixed) type. The alternative entropy framework that is proposed here is based on a two step approach. In the first step, infinitely many classes of entropy solutions are defined, each associated with an interface connection. We show that each of these class of entropy solutions form a contractive semigroup in L1 and is hence unique. Godunov type schemes based on solutions of the Riemann problem are designed and shown to converge to each class of these entropy solutions. The second step is to choose one of these classes of solutions. This choice depends on the Physics of the problem being considered and we concentrate on the model of two-phase flows in a heterogeneous porous medium. We define an optimization problem on the set of admissible interface connections and show the existence of an unique optimal connection and its corresponding optimal entropy solution. The optimal entropy solution is consistent with the expected solutions for two-phase flows in heterogeneous porous media.

I and i

Abstract: There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality. Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

Linear and nonlinear waves, by G. B. Whitham. Pp.636. £50. 1999. ISBN 0 471 35942 4 (Wiley).

Abstract: to be done in this area. Face recognition is a problem that scarcely clamoured for attention before the computer age but, having surfaced, has involved a wide range of techniques and has attracted the attention of some fine minds (David Mumford was a Fields Medallist in 1974). This singular application of mathematical modelling to a messy applied problem of obvious utility and importance but with no unique solution is a pretty one to share with students: perhaps, returning to the source of our opening quotation, we may invert Duncan's earlier observation, 'There is an art to find the mind's construction in the face!'.

The soil production function and landscape equilibrium

Abstract: ). The linear increase of detected photons as a function of laser intensity (100-2,000 W cm -2 ) indicated that saturation and multiphoton processes were negligible in these studies. Typical detected count rates of 5,000-6,000 photons s -1 at 2,000 W cm -2 pumping intensity (,150,000 excitations s -1 ) were achieved, with most of the molecules emitting several

Nonlinear partial differential equations with applications

Abstract: Preface.- Preface to the 2nd edition.- Notational conventions.- 1 Preliminary general material.- I Steady-state problems.- 2 Pseudomonotone or weakly continuous mappings.- 3 Accretive mappings.- 4 Potential problems: smooth case.- 5 Nonsmooth problems variational inequalities.- 6. Systems of equations: particular examples.- II Evolution problems.- 7 Special auxiliary tools.- 8 Evolution by pseudomonotone or weakly continuous mappings.- 9 Evolution governed by accretive mappings.- 10 Evolution governed by certain set-valued mappings.- 11 Doubly-nonlinear problems.- 12 Systems of equations: particular examples.- References.- Index.

A Sharp Trudinger-Moser Type Inequality for Unbounded Domains in Rn

Abstract: The classical Trudinger–Moser inequality says that for functions with Dirichlet norm smaller or equal to 1 in the Sobolev space H 0 1 ( Ω ) (with Ω ⊂ R 2 a bounded domain), the integral ∫ Ω e 4 π u 2 dx is uniformly bounded by a constant depending only on Ω . If the volume | Ω | becomes unbounded then this bound tends to infinity, and hence the Trudinger–Moser inequality is not available for such domains (and in particular for R 2 ). In this paper, we show that if the Dirichlet norm is replaced by the standard Sobolev norm, then the supremum of ∫ Ω e 4 π u 2 dx over all such functions is uniformly bounded, independently of the domain Ω . Furthermore, a sharp upper bound for the limits of Sobolev normalized concentrating sequences is proved for Ω = B R , the ball or radius R, and for Ω = R 2 . Finally, the explicit construction of optimal concentrating sequences allows to prove that the above supremum is attained on balls B R ⊂ R 2 and on R 2 .