Satyanad Kichenassamy

On a conjecture of Fefferman and Graham

Abstract In 1985, Fefferman and Graham have constructed a local embedding of an arbitrary real-analytic manifold, of odd dimension n , into a Ricci-flat manifold of dimension n +2 admitting a homothety. They conjectured that their result remains valid in even dimensions, if logarithms are allowed in the expansion of the metric. In this paper, we (i) prove that such expansions exist and converge and (ii) establish the degree of non-uniqueness of the solution in terms of the coefficients in the expansion.

A linear Fuchsian equation with variable indices

Abstract We construct three types of solutions for a Fuchsian equation with variable indices: (1) branched solutions involving logarithms of the time variable t; (2) solutions involving tx, where x is a space variable; and, (3) for a model case, exact solutions involving hypergeometric functions. These three solutions have completely different singularities. The constructions are given in a form suitable for application to more general equations. As an illustration, we resolve in particular an apparent discrepancy between two recent results on this problem.

Soliton stars in the breather limit

This paper presents an asymptotic reduction of the Einstein–Klein–Gordon system with a real scalar field (soliton star problem). A periodic solution of the reduced system, similar to the sine-Gordon breather, is obtained by a variational method. This tallies with numerical computations. As a consequence, a time-periodic redshift for sources close to the center of the star is obtained.

HYPERGEOMETRIC FUNCTIONS AND SINGULAR SOLUTIONS OF WAVE EQUATIONS WITH LORENTZ-INVARIANT POTENTIAL

New solutions of equation utt - Δu + Au/(r2 - t2) = 0, in the series of hypergeometric functions (HGF), are constructed. A sharpened estimate on hypergeometric functions is derived to prove the convergence of the series.

Recent Progress on Boundary Blow-up

We report on the solution of two long-standing conjectures on the boundary behavior of maximal solutions of semilinear elliptic equations, focusing on the proof of the boundary regularity of the hyperbolic radius in higher dimensions. The main tool is the reduction of the problem to a degenerate equation of Fuchsian type, for which new Schauder-type estimates are proved. We also sketch an algorithm suitable for large classes of applications.

Explosion et normes Lp pour l'équation des ondes nonlinéaire cubique

We find blow-up solutions of nonlinear wave equations with cubic nonlinearity, in any number of space dimensions, and study the asymptotic behavior of their Lp norms and “energy”. The Lp norm blows up if the blow-up surface has an interior non-degenerate minimum and p⩾n/2. For less smooth right-hand sides, and 0<e<1, we give examples for which the Lp norm blows up if p⩾n/(1+e); their Cauchy data are unbounded, but blow-up is not instantaneous. Applications to nonlinear optics are briefly outlined. To cite this article: G. Cabart, S. Kichenassamy, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 903–908.

Chapter 5 Schauder-type estimates and applications

Publisher Summary 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 Holder regularity estimates for solutions of elliptic problems with Holder 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 Cauchys 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.