Hölder regularity for a Kolmogorov equation

We study the interior regularity properties of the solutions to the degenerate parabolic equation,

which arises in mathematical finance and in the theory of diffusion processes.

     

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 .

     

Regularity of a free boundary in parabolic potential theory

We study the regularity of the free boundary in a Stefan-type problem

with no sign assumptions on and the time derivative .

     

Solvability of a finite or infinite system of discontinuous quasimonotone differential equations

This paper proves the existence of solutions to the initial value problem

where may be discontinuous but is assumed to satisfy conditions of superposition-measurability, quasimonotonicity, quasisemicontinuity, and integrability. The set can be arbitrarily large (finite or infinite); our theorem is new even for . The proof is based partly on measure-theoretic techniques used in one dimension under slightly stronger hypotheses by Rzymowski and Walachowski. Further generalizations are mentioned at the end of the paper.

     

Strongly indefinite functionals and multiple solutions of elliptic systems

We study existence and multiplicity of solutions of the elliptic system

where , is a smooth bounded domain and . We assume that the nonlinear term

where , , and . So some supercritical systems are included. Nontrivial solutions are obtained. When is even in , we show that the system possesses a sequence of solutions associated with a sequence of positive energies (resp. negative energies) going toward infinity (resp. zero) if 2$"> (resp. ). All results are proved using variational methods. Some new critical point theorems for strongly indefinite functionals are proved.

     

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.

     

Exact multiplicity result for the perturbed scalar curvature problem in

Let denote the closure of in the norm Let and define the constants and Let We consider the following problem for

We show an exact multiplicity result for for all small .

     

Existence of positive solutions for a semilinear elliptic problem with critical Sobolev and Hardy terms

Let , let and let be a bounded domain with a smooth boundary . Our purpose in this paper is to consider the existence of solutions of the problem:

where

     

Multiple symmetric positive solutions of a class of boundary value problems for higher order ordinary differential equations

In this paper, the authors consider the boundary value problem

and give sufficient conditions for the existence of any number of symmetric positive solutions of (E)-(B). The relationships between the results in this paper and some recent work by Henderson and Thompson (Proc. Amer. Math. Soc. 128 (2000), 2373-2379) are discussed.

     

The linear heat equation with highly oscillating potential

In this paper we consider the following initial value problem:

where and . Nonexistence of positive solutions is analyzed.

     

Geometry of phase space and solutions of semilinear elliptic equations in a ball

We consider the problem

where denotes the unit ball in , , and . Merle and Peletier showed that for there is a unique value such that a radial singular solution exists. This value is the only one at which an unbounded sequence of classical solutions of (1) may accumulate. Here we prove that if additionally

then for close to , a large number of classical solutions of (1) exist. In particular infinitely many solutions are present if . We establish a similar assertion for the problem

where , , and satisfies the same condition as above.

     

Semiclassical limits of ground state solutions to Schrödinger systems

This paper is concerned with the existence and concentration properties of the ground state solutions to the following coupled Schrödinger systems $$\begin{aligned} \left\{ \begin{array}{l} -\varepsilon ^2\varDelta u+u+V(x)v=W(x)G_{v}(z)~\hbox { in }\ {\mathbb {R}}^N,\\ -\varepsilon ^2\varDelta v+v+V(x)u=W(x)G_{u}(z)~\hbox {in } \ {\mathbb {R}}^N,\\ u(x)\rightarrow 0\ \hbox {and }v(x)\rightarrow 0\ \hbox {as } \ |x|\rightarrow \infty , \end{array} \right. \end{aligned}$$ and $$\begin{aligned} \left\{ \begin{array}{l} -\varepsilon ^2\varDelta u+u+V(x)v=W(x)(G_{v}(z)+|z|^{2^*-2}v)~\hbox {in } \ {\mathbb {R}}^N,\\ -\varepsilon ^2\varDelta v+v+V(x)u=W(x)(G_{u}(z)+|z|^{2^*-2}u)~\hbox {in } \ {\mathbb {R}}^N,\\ u(x)\rightarrow 0\ \hbox {and }v(x)\rightarrow 0\ \hbox {as } \ |x|\rightarrow \infty , \end{array} \right. \end{aligned}$$ where \(z=(u,v)\in {\mathbb {R}}^2\) , \(G\) is a power type nonlinearity, having superquadratic growth at both \(0\) and infinity but subcritical, \(V\) can be sign-changing and \(\inf W>0\) . We prove the existence, exponential decay, \(H^2\) -convergence and concentration phenomena of the ground state solutions for small \(\varepsilon >0\) .     

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

     

Viscosity and relaxation approximations for a hyperbolic-elliptic mixed type system

To a given system of conservation laws

we associate the system

which is of mixed type. Under certain conditions, convergence of this latter system for with is established without the need of stability criteria or hyperbolicity of the left-hand sides of the equations.

     

The quartile operator and pointwise convergence of Walsh series

The bilinear Hilbert transform is given by

It satisfies estimates of the type

In this paper we prove such estimates for a discrete model of the bilinear Hilbert transform involving the Walsh Fourier transform. We also reprove the well-known fact that the Walsh Fourier series of a function in , with converges pointwise almost everywhere. The purpose of this exposition is to clarify the connection between these two results and to present an easy approach to recent methods of time-frequency analysis.

     

Quasiconvexity and uniqueness of stationary points in the multi-dimensional calculus of variations

Let be a bounded starshaped domain. In this note we consider critical points of the functional

where of class satisfies the natural growth

for some and 0$">, is suitably rank-one convex and in addition is strictly quasiconvex at . We establish uniqueness results under the extra assumption that is stationary at with respect to variations of the domain. These statements should be compared to the uniqueness result of Knops & Stuart (1984) in the smooth case and recent counterexamples to regularity produced by Müller & Sverák (2003).

     

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.$">

     

Twisted higher moments of Kloosterman sums

Let be a nontrivial Dirichlet character modulo an odd prime . Write

We shall prove

and, for complex ,

0, \end{displaymath}">

where is a constant depending only on .

     

A criterion for correct solvability of the Sturm-Liouville equation in the space

We consider an equation

where , and By a solution of equation (1), we mean any function such that and equality (1) holds almost everywhere on In this paper, we obtain a criterion for the correct solvability of (1) in ,

     

On positive periodic solutions of Lotka-Volterra competition systems with deviating arguments

By using Krasnoselskii's fixed point theorem, we prove that the following periodic species Lotka-Volterra competition system with multiple deviating arguments

has at least one positive periodic solution provided that the corresponding system of linear equations

has a positive solution, where and are periodic functions with

Furthermore, when and , , are constants but , remain -periodic, we show that the condition on is also necessary for to have at least one positive periodic solution.