arXiv subjects

Satyanad Kichenassamy

Publications and source records attributed to Satyanad Kichenassamy.

At least 19 recordsLinked to original sources

arXiv2025

Symmetrization on the sphere and applications

We introduce a new method of symmetrization of mappings on the $n$-sphere ($n\geq 2$). They are applied to estimate solutions of quasilinear elliptic partial differential equations of $p$-Laplacian type, with combinations of Dirac measures on the right-hand side. The case $p=n$ is reduced to a problem on the sphere, using a conformal transformation. The cases when $1 < p < n $ and $p > $n are considered more briefly, full details being available in other papers of the author.

math.AP↗

arXiv2025

Symmetric hyperbolic systems and shock waves

The theory of symmetric-hyperbolic systems is useful for constructing smooth solutions of nonlinear wave equations, and for studying their singularities, including shock waves. We present the main techniques which are required to apply the theory in Mathematical Physics.

math.AP↗

arXiv2025

Wave Mechanics and C-equivalence

We establish that de Broglie's wave, as it is introduced in his Thesis, is not a wave on a given space, but on the contrary itself determines a system in which Special Relativity is locally valid. This local system is a physical object, that defines its own units of length and time. C-equivalence provides a natural framework both for its mathematical description and its physical interpretation.

physics.hist-ph↗

arXiv2025

Perimeter on a manifold, with applications to partial differential equations

The perimeter of a measurable subset of $\mathbb R^N$ is the total variation of its characteristic function. We generalize this notion to a subset $E$ of a closed Riemannian manifold. We show that the perimeter of $E$ is the limit of the hear kernel regularization of its characteristic function. A generalization of the isoperimetric inequality and of the Fleming-Rishel formula follow. These results are applied to a quasilinear elliptic problem in $\mathbb R^N$ for which the usual symmetrization methods fail. It will be tackled successfully by introducing a symmetrization method on the sphere.

math.AP↗

arXiv2025

Textual analysis of ancient Indian mathematics

Recent analyses of Brahmagupta's discourse on the cyclic quadrilateral, and of Baudhāyana's approximate quadrature of the circle, have shown that it is useful to submit mathematical texts to a form of literary analysis. Several passages considered as obscure or objectionable may be explained in this way, by taking into account the elements of exposition and derivation of the results that the author has given, as well as his conceptual background. This approach aims at helping the reader set aside his preconceptions about what a mathematical text is supposed to be. In this paper, guidelines for further application of this method are outlined, with illustrations taken from our previous papers.

math.HO↗

arXiv2025

Régularité du rayon hyperbolique

Let $Ω\subset\mathbb R^2$ be a bounded domain of class $C^{2+α}$, $0<α<1$. We show that if $u$ is the maximal solution of $Δu = 4\exp(2u)$, which tends to $+\infty$ as $(x,y)\to\partialΩ$, then the hyperbolic radius $v=\exp(-u)$ is of class $C^{2+α}$ up to the boundary. The proof relies on new Schauder estimates for Fuchsian elliptic equations.

math.AP↗

arXiv2025

Boundary behavior in the Loewner-Nirenberg problem

Let $Ω\subset\mathbb R^n$ be a bounded domain of class $C^{2+α}$, $0<α<1$. We show that if $n\geq 3$ and $u_Ω$ is the maximal solution of equation $Δu = n(n-2)u^{(n+2)/(n-2)}$ in $Ω$, then the hyperbolic radius $v_Ω=u_Ω^{-2/(n-2)}$ is of class $C^{2+α}$ up to the boundary. The argument rests on a reduction to a nonlinear Fuchsian elliptic PDE.

math.CV↗

arXiv2025

Boundary blow-up and degenerate equations

Let $Ω\subset\mathbb R^2$ be a bounded domain of class $C^{2+α}$, $0<α<1$. We show that if $u$ is the solution of $Δu = 4\exp(2u)$ which tends to $+\infty$ as $(x,y)\to\partialΩ$, then the hyperbolic radius $v=\exp(-u)$ is also of class $C^{2+α}$ up to the boundary. The proof relies on new Schauder estimates for degenerate elliptic equations of Fuchsian type.

math.CV↗

arXiv2025

Renormalized variational principles and Hardy-type inequalities

Let $Ω\subset{\mathbb R}^2$ be a bounded domain on which Hardy's inequality holds. We prove that $[\exp(u^2)-1]/δ^2\in L^1(Ω)$ if $u\in H^1_0(Ω)$, where $δ$ denotes the distance to $\partialΩ$. The corresponding higher-dimensional result is also given. These results contain both Hardy's and Trudinger's inequalities, and yield a new variational characterization of the maximal solution of the Liouville equation on smooth domains, in terms of a renormalized functional. A global $H^1$ bound on the difference between the maximal solution and the first term of its asymptotic expansion follows.

math.AP↗

arXiv2025

Schauder-type estimates and applications

The Schauder estimates are among the oldest and most useful tools in the modern theory of elliptic partial differential equations (PDEs). Their influence may be felt in practically all applications of the theory of elliptic boundary-value problems, that is, in fields such as nonlinear diffusion, potential theory, field theory or differential geometry and its applications. Schauder estimates give Hölder regularity estimates for solutions of elliptic problems with Hölder continuous data; they may be thought of as wide-ranging generalizations of estimates of derivatives of an analytic function in the interior of its domain of analyticity and play a role comparable to that of Cauchy's theory in function theory. They may be viewed as converses to the mean-value theorem: a bound on the solution gives a bound on its derivatives. Schauder theory has strongly contributed to the modern idea that solving a PDE is equivalent to obtaining an a priori bound that is, trying to estimate a solution before any solution has been constructed. The chapter presents the complete proofs of the most commonly used theorems used in actual applications of the estimates.

math.AP↗

arXiv2025

Further examples of apodictic discourse, I

The analysis of problematic mathematical texts, particularly from India, has required the introduction of a new category of rigorous discourse, apodictic discourse. We briefly recall why this introduction was necessary. We then show that this form of discourse is widespread among scholars, even in contemporary Mathematics, in India and elsewhere. It is in India a natural outgrowth of the emphasis on non-written communication, combined with the need for freedom of thought. New results in this first part include the following: (i) {Ā}ryabha___a proposed a geometric derivation of a basic algebraic identity; (ii) Brahmagupta proposed an original argument for the irrationality of quadratic surds on the basis of his results on the varga-prak___ti problem, thereby justifying his change in the definition of the word kara___{ī}.

math.HO↗

arXiv2025

Apodictic discourse and the Cauchy-Bunyakovsky-Schwarz inequality

Bunyakovsky's integral inequality (1859) is one of the familiar tools of modern Analysis. We try and understand what Bunyakovsky did, why he did it, why others did not follow the same path, and explore some of the mathematical (re)interpretations of his inequalities. This is achieved by treating the texts as discourses that provide motivation and proofs by their very discursive structure, in addition to what meets the eye at first reading. Bunyakovsky paper is an outgrowth of the mathematical theory of mean-values in Cauchy's work (1821), but viewed from the point of view of Probability and Statistics. Liouville (1836) gave a result that implies Bunyakovsky's inequality, but did not identify it as significant because his interests lay elsewhere. Grassmann (1862) stated the inequality in abstract form but did not prove it for reasons that can be identified. Finally, by relating the result to quadratic binary forms, Schwarz (1885) opened the way to a geometric interpretation of the inequality that became important in the theory of integral equations. His argument is the source of one of the proofs most commonly taught nowadays. At about the same time, the Rogers-H{ö}lder inequality suggested generalizations of Cauchy's and Bunyakovsky's results in an entirely different direction. Later extensions and reinterpretations show that no single result, even now, subsumes all known generalizations.

math.AP↗

arXiv2024

The Relativistic Rotation Transformation and the Observer Manifold

We show that relativistic rotation transformations represent transfer maps between the laboratory system and a local observer on an observer manifold, rather than an event manifold, in the spirit of C-equivalence. Rotation is, therefore, not a parameterised motion on a background space or spacetime, but is determined by a particular sequence of tetrads related by specific special Lorentz transformations or boosts. Because such Lorentz boosts do not form a group, these tetrads represent distinct observers that cannot put together their local descriptions into a manifold in the usual sense. The choice of observer manifold depends on the dynamical situation under consideration, and is not solely determined by the kinematics. Three examples are given: Franklin's rotation transformation for uniform plane rotation, the Thomas precession of a vector attached to an electron, and the motion of a charged particle in an electromagnetic field. In each case, at each point of its trajectory, there is a distinguished tetrad and a special Lorentz transformation that maps Minkowski space to the spacetime of the local observer on the curve.

math.GM↗

arXiv2022

Introduction {à} l'oeuvre de S. Kichenassamy en Physique Th{é}orique

The work of S. Kichenassamy (1926-2015) in Theoretical and Mathematical Physics covers the spectrum of Relativistic Physics: from the clarification of the postulational basis of the two theories of Relativity, to applications to the measurement of proper time, image formation, collision theory, kinetic theory and radiative transfer, or pulsar electrodynamics, to name a few. After brief biographic remarks,we show, by following an argument first formulated in 1963, and developed in a series of papers, that the physical interpretation of General Relativity requires the replacement of the strong equivalence principle by the C-equivalence principle. By giving a mathematical status to the observer, his insight yields new results regarding measurements actual observers can make, both in Special and in General Relativity. Applications are outlined. His contributions to Indology will be discussed elsewhere.

physics.hist-ph↗

arXiv2022

Instability of pole singularities for the Chazy equation

We prove that the negative resonances of the Chazy equation (in thesense of Painlevé analysis) can be related directly to it sgroup-invariance properties. These resonances indicate in this case the instability of pole singularities. Depending on the value of a parameter in the equation, an unstable isolated pole may turn into the familiar natural boundary, or split into several isolated singularities. In the first case, a convergent series representation involving exponentially small corrections can be given. This reconciles several earlier approaches to the interpretation of negative resonances. On the other hand, we also prove that pole singularities with the maximum number of positive resonances are stable. The proofs rely on general properties of nonlinear Fuchsian equations.

nlin.SI↗

arXiv2017

Analytic description of singularities in Gowdy spacetimes

We use Fuchsian Reduction to construct singular solutions of Einstein's equations which belong to the class of Gowdy spacetimes. The solutions have the maximum number of arbitrary functions. Special cases correspond to polarized, or other known solutions. The method provides precise asymptotics at the singularity, which is Kasner-like. All of these solutions are asymptotically velocity-dominated. The results account for the fact that solutions with velocity parameter uniformly greater than one are not observed numerically. They also provide a justification of formal expansions proposed by Grubišić and Moncrief.

gr-qc↗

arXiv2017

A study of ancient Khmer ephemerides

We study ancient Khmer ephemerides described in 1910 by the French engineer Faraut, in order to determine whether they rely on observations carried out in Cambodia. These ephemerides were found to be of Indian origin and have been adapted for another longitude, most likely in Burma. A method for estimating the date and place where the ephemerides were developed or adapted is described and applied.

physics.hist-ph↗

arXiv2017

Asymptotic Behavior in Polarized {\bf T}$^2$-symmetric Vacuum Spacetimes

We use Fuchsian Reduction to study the behavior near the singularity of a class of solutions of Einstein's vacuum equations. These solutions admit two commuting spacelike Killing fields like the Gowdy spacetimes, but their twist does not vanish. The spacetimes are also polarized in the sense that one of the `gravitational degrees of freedom' is turned off. Examining an analytic family of solutions with the maximum number of arbitrary functions, we find that they are all asymptotically velocity-term dominated as one approaches the singularity.

gr-qc↗