Gaussian estimates for fundamental solutions of second order parabolic systems with time-independent coefficients

Auscher, McIntosh and Tchamitchian studied the heat kernels of second order elliptic operators in divergence form with complex bounded measurable coefficients on . In particular, in the case when they obtained Gaussian upper bound estimates for the heat kernel without imposing further assumption on the coefficients. We study the fundamental solutions of the systems of second order parabolic equations in the divergence form with bounded, measurable, time-independent coefficients, and extend their results to the systems of parabolic equations.

     

Tornado solutions for semilinear elliptic equations in : Applications

We show partial regularity of bounded positive solutions of some semilinear elliptic equations in domains of . As a consequence, there exists a large variety of nonnegative singular solutions to these equations. These equations have previously been studied from the point of view of free boundary problems, where solutions additionally are stable for a variational problem, which we do not assume.

     

Almost optimal local well-posedness for the (3+1)-dimensional Maxwell-Klein-Gordon equations

We prove that the evolution problem for the Maxwell-Klein-Gordon system is locally well posed when the initial data belong to the Sobolev space for any 0$">. This is in spite of a complete failure of the standard model equations in the range . The device that enables us to obtain inductive estimates is a new null structure which involves cancellations between the elliptic and hyperbolic terms in the full equations.

     

Counting integral Lamé equations by means of dessins d'enfants

We obtain an explicit formula for the number of Lamé equations (modulo linear changes of variable) with index and projective monodromy group of order , for given and . This is done by performing the combinatorics of the `dessins d'enfants' associated to the Belyi covers which transform hypergeometric equations into Lamé equations by pull-back.

     

On polynomial-factorial diophantine equations

We study equations of the form and show that for some classes of polynomials the equation has only finitely many solutions. This is the case, say, if is irreducible (of degree greater than 1) or has an irreducible factor of ``relatively large" degree. This is also the case if the factorization of contains some ``large" power(s) of irreducible(s). For example, we can show that the equation has only finitely many solutions for , but not that this is the case for (although it undoubtedly should be). We also study the equation , where is one of several other ``highly divisible" sequences, proving again that for various classes of polynomials these equations have only finitely many solutions.

     

Tornado solutions for semilinear elliptic equations in : regularity

We give conditions under which bounded solutions to semilinear elliptic equations on domains of are continuous despite a possible infinite singularity of . The conditions do not require a minimization or variational stability property for the solutions. The results are used in a second paper to show regularity for a familiar class of equations.

     

Modular differential equations of second order with regular singularities at elliptic points for

We give a definition of the modular differential equations of weight for a discrete subgroup for ; in this paper we set . We solve such equations admitting regular singularities at elliptic points for in terms of the Eisenstein series and the Gauss hypergeometric series. Furthermore, we give a series of such modular differential equations parametrized by an even integer , and discuss some properties of solution spaces. We find several equations which are solved by a modular form of weight .

     

More limit cycles than expected in Liénard equations

The paper deals with classical polynomial Liénard equations, i.e. planar vector fields associated to scalar second order differential equations where is a polynomial. We prove that for a well-chosen polynomial of degree the equation exhibits limit cycles. It induces that for there exist polynomials of degree such that the related equations exhibit more than limit cycles. This contradicts the conjecture of Lins, de Melo and Pugh stating that for Liénard equations as above, with of degree the maximum number of limit cycles is The limit cycles that we found are relaxation oscillations which appear in slow-fast systems at the boundary of classical polynomial Liénard equations. More precisely we find our example inside a family of second order differential equations Here, is a well-chosen family of polynomials of degree with parameter and is a small positive parameter tending to We use bifurcations from canard cycles which occur when two extrema of the critical curve of the layer equation are crossing (the layer equation corresponds to . As was proved by Dumortier and Roussarie (2005) these bifurcations are controlled by a rational integral computed along the critical curve of the layer equation, called the slow divergence integral. Our result is deduced from the study of this integral.

     

Projective sets and ordinary differential equations

We prove that for the set of Cauchy problems of dimension which have a global solution is -complete and that the set of ordinary differential equations which have a global solution for every initial condition is -complete. The first result still holds if we restrict ourselves to second order equations (in dimension one). We also prove that for the set of Cauchy problems of dimension which have a global solution even if we perturb a bit the initial condition is -complete.

     

Numerical index and renorming

We study the numerical index of a Banach space from the isomorphic point of view, that is, we investigate the values of the numerical index which can be obtained by renorming the space. The set of these values is always an interval which contains in the real case and in the complex case. Moreover, for ``most' Banach spaces the least upper bound of this interval is as large as possible, namely .