Sharp maximal estimates for doubly oscillatory integrals

We study doubly oscillatory integrals

and prove a sharp maximal estimate which is an immediate consequence of a well-known conjecture in Fourier analysis on .

     

On finite-time blow-up for a nonlocal parabolic problem arising from shear bands in metals

Results on finite-time blow-up of solutions to the nonlocal parabolic problem

are established. They extend some known results to higher dimensions.

     

On the strong maximum principle

This paper presents a necessary and sufficient condition on the convex function in order that continuous solutions to

satisfy a Strong Maximum Principle on any open connected .

     

On the evaluation of generalized Watson integrals

The triple integrals

and

where and are complex variables in suitably defined cut planes, were first evaluated by Watson in 1939 for the special cases and , respectively. In the present paper simple direct methods are used to prove that can be expressed in terms of squares of complete elliptic integrals of the first kind for general values of and . It is also shown that and are related by the transformation formula

where

Thus both of Watson's results for are contained within a single formula for .

     

On the Well-Posedness of the Kirchhoff String

Let us consider the Cauchy problem for the quasilinear hyperbolic integro-differential equation

where is an open subset of and is a positive function of one real variable which is continuously differentiable. We prove the well-posedness in the Hadamard sense (existence, uniqueness and continuous dependence of the local solution upon the initial data) in Sobolev spaces of low order.

     

The Laplace transform of the digamma function: An integral due to Glasser, Manna and Oloa

The definite integral

is related to the Laplace transform of the digamma function

by when . Certain analytic expressions for in the complementary range, , are also provided.

     

Irrational rotation numbers and unboundedness of solutions of the second order differential equations with asymmetric nonlinearities

In this paper, we study the dynamics of the mappings

where is a irrational rotation number. We prove the existence of orbits that go to infinity in the future or in the past by using the well-known Birkhoff Ergodic Theorem. Applying this conclusion, we deal with the unboundedness of solutions of Liénard equations with asymmetric nonlinearities.

     

On a Liouville-type theorem and the Fujita blow-up phenomenon

The main purpose of this paper is to obtain the well-known results of H.Fujita and K.Hayakawa on the nonexistence of nontrivial nonnegative global solutions for the Cauchy problem for the equation

with on the half-space as a consequence of a new Liouville theorem of elliptic type for solutions of () on . This new result is in turn a consequence of other new phenomena established for nonlinear evolution problems. In particular, we prove that the inequality

has no nontrivial solutions on when We also show that the inequality

has no nontrivial nonnegative solutions for , and it has no solutions on bounded below by a positive constant for 1.$">

     

Lack of natural weighted estimates for some singular integral operators

We show that the classical Hörmander condition, or analogously the -Hörmander condition, for singular integral operators is not sufficient to derive Coifman's inequality

where , is the Hardy-Littlewood maximal operator, is any weight and is a constant depending upon and the constant of . This estimate is well known to hold when is a Calderón-Zygmund operator.

As a consequence we deduce that the following estimate does not hold:

where and where is an arbitrary weight. However, by a recent result due to A. Lerner, this inequality is satisfied whenever is a Calderón-Zygmund operator.

One of the main ingredients of the proof is a very general extrapolation theorem for weights.

     

An estimate for the number of bound states of the Schrödinger operator in two dimensions

For the Schrödinger operator on let be the number of bound states. One obtains the following estimate:

where and ( is the Euler constant). This estimate holds for all potentials for which the previous integral is finite.

     

A NECESSARY AND SUFFICIENT CONDITION OF EXISTENCE OF GLOBAL SOLUTIONS FOR SOME NONLINEAR HYPERBOLIC EQUATIONS

ＡＮＥＣＥＳＳＡＲＹＡＮＤＳＵＦＦＩＣＩＥＮＴＣＯＮＤＩＴＩＯＮＯＦＥＸＩＳＴＥＮＣＥＯＦＧＬＯＢＡＬＳＯＬＵＴＩＯＮＳＦＯＲＳＯＭＥＮＯＮＬＩＮＥＡＲＨＹＰＥＲＢＯＬＩＣＥＱＵＡＴＩＯＮＳ￥ＺＨＡＮＧＱＵＡＮＤＥ（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅ...     

On a Littlewood-Paley type inequality

The following is proved: If is a function harmonic in the unit ball and if then the inequality

holds, where is the nontangential maximal function of This improves a recent result of Stoll. This inequality holds for polyharmonic and hyperbolically harmonic functions as well.

     

A short proof of an inequality of Littlewood and Paley

A very short proof is given of the inequality

where and is the Poisson integral of

     

On the Bochner theorem on orthogonal operators

We prove the following theorem. Any isometric operator , that acts from the Hilbert space with nonnegative weight to the Hilbert space with nonnegative weight , allows for the integral representation

where the kernels and satisfy certain conditions that are necessary and sufficient for these kernels to generate the corresponding isometric operators.

     

Forced oscillation of second order linear and half-linear difference equations

Oscillation properties of solutions of the forced second order linear difference equation

are investigated. The authors show that if the forcing term does not oscillate, in some sense, too rapidly, then the oscillation of the unforced equation implies oscillation of the forced equation. Some results illustrating this statement and extensions to the more general half-linear equation

1, \end{displaymath}">

are also given.

     

Integrals of polylogarithmic functions, recurrence relations, and associated Euler sums

We show that integrals of the form

and

satisfy certain recurrence relations which allow us to write them in terms of Euler sums. From this we prove that, in the first case for all and in the second case when is even, these integrals are reducible to zeta values. In the case of odd , we combine the known results for Euler sums with the information obtained from the problem in this form to give an estimate on the number of new constants which are needed to express the above integrals for a given weight .

The proofs are constructive, giving a method for the evaluation of these and other similar integrals, and we present a selection of explicit evaluations in the last section.

     

Representation formulae and inequalities for solutions of a class of second order partial differential equations

Let be a possibly degenerate second order differential operator and let be its fundamental solution at ; here is a suitable distance. In this paper we study necessary and sufficient conditions for the weak solutions of on to satisfy the representation formula

We prove that (R) holds provided is superlinear, without any assumption on the behavior of at infinity. On the other hand, if satisfies the condition

then (R) holds with no growth assumptions on .

     

Symmetry-Breaking Phenomena in an Optimization Problem for some Nonlinear Elliptic Equation

Let $\Omega$ be a bounded domain in ${\bf R^n}$ with Lipschitz boundary, $\lambda >0,$ and $1\le p \le (n+2)/(n-2)$ if $n\ge 3$ and $1\le p< +\infty$ if $n=1,2$. Let $D$ be a measurable subset of $\Omega$ which belongs to the class $ {\cal C}_{\beta}=\{D\subset \Omega \quad | \quad |D|=\beta\} $ for the prescribed $\beta\in (0, |\Omega|).$ For any $D\in{\cal C}_{\beta}$, it is well known that there exists a unique global minimizer $u\in H^1_0(\Omega)$, which we denote by $u_D$, of the functional \[\quad J_{\Omega,D}(v)=\frac12\int_{\Omega}|\nabla v|^2\, dx+\frac{\lambda}{p+1}\int_{\Omega}|v|^{p+1}\, dx -\int_{\Omega}\chi_Dv\,dx \] on $H^1_0(\Omega)$. We consider the optimization problem $ E_{\beta,\Omega}=\inf_{D\in {\cal C}_{\beta}} J_D(u_D) $ and say that a subset $D^*\in {\cal C}_{\beta}$ which attains $E_{\beta,\Omega}$ is an optimal configuration to this problem. In this paper we show the existence, uniqueness and non-uniqueness, and symmetry-preserving and symmetry-breaking phenomena of the optimal configuration $D^*$ to this optimization problem in various settings.     

Functorial equations for lexicographic products

We generalize the main result of an earlier paper by the authors (Exponentiation in power series fields, Proc. Amer. Math. Soc. 125 (1997), 3177-3183) concerning the convex embeddings of a chain in a lexicographic power . For a fixed non-empty chain , we derive necessary and sufficient conditions for the existence of non-empty solutions to each of the lexicographic functorial equations

, (\Delta ^{\Gam... ...Gamma \mbox{ and } (\Delta ^{\Gamma})^{<0} \simeq \Gamma\;.\end{displaymath}">

     

Singular integral operators associated to curves with rational components

We prove , bounds for

and

where , are rational functions. Our bounds depend only on the degrees of the polynomials and, in particular, they do not depend on the coefficients of these polynomials.