Estimates for the Cauchy function of linear singular differential equations and some applications

For higher-order linear singular equations, we find sharp estimates for the Cauchy function and its partial derivatives. We use these estimates to study the properties of solutions of singular differential inequalities and obtain optimal sufficient conditions for the unique solvability of the linear singular Cauchy problem.     

Existence and uniqueness of weak solutions for some singular evolutionary system in image denoising

In this paper, we prove the existence and uniqueness of weak solutions for some singular evolutionary system, which is deduced from one of the Chambolle–Lions denoising models. The main method we used is p‐Laplace regularization. Copyright © 2013 John Wiley & Sons, Ltd.     

On some properties of singular matrices

Some results of Ostrowski in [5] are generalized to the case of monotonic norms.     

On the monotonicity of some singular integral operators

The paper deals with the monotonicity of singular integral operators of the form where is the Cauchy singular integral operator on the interval (0,1) of the real axis and q is a power or logarithmic function. Under suitable assumptions, such singular integral operators are proved to be monotone and maximal monotone in spaces with power weights. Moreover, two related integral equations with weakly singular kernels of logarithmic type are studied. Copyright © 2012 John Wiley & Sons, Ltd.     

Solutions of some singular integral equation

A singular integral equation with a Holderian second member function on [a,b] is considered and solved for four different type of kernels in the class of functions that are unbounded at the end points of the interval.     

Elliptic system with some singular nonlinearity

We investigate multiplicity of solutions for elliptic systems with singular non-linearities. We obtain a theorem which shows that elliptic systems with some singular non-linearities have infinitely many solutions. We get this result by using the variational method, critical point theory and homology theory.     

On some singular limits of homogeneous semigroups

No Abstract. RID="h1" ID="h1"In memory of our friend Philippe Bénilan?“Mais là où les uns voyaient l'abstraction d'autres voyaient la vérité”?Camus, La Peste     

Essential norms of some singular integral operators

Let a\alpha and b\beta be bounded measurable functions on the unit circle T. The singular integral operator S_a, bS_{\alpha ,\,\beta } is defined by S_a, b f = aPf + bQf(f ? L² (T))S_{\alpha ,\,\beta } f = \alpha Pf + \beta Qf(f \in L^2 (T)) where P is an analytic projection and Q is a co-analytic projection. In the previous paper, the norm of S_a, bS_{\alpha ,\,\beta } was calculated in general, using a,b\alpha ,\beta and a[`(b)] + H<sup>￥</sup>\alpha \bar {\beta } + H^\infty where H<sup>￥</sup>H^\infty is a Hardy space in L<sup>￥</sup> (T).L^\infty (T). In this paper, the essential norm ||S<sub>a, b</sub> ||<sub>e</sub>\Vert S_{\alpha ,\,\beta } \Vert _e of S<sub>a, b</sub>S_{\alpha ,\,\beta } is calculated in general, using a[`(b)] + H^￥ + C\alpha \bar {\beta } + H^\infty + C where C is a set of all continuous functions on T. Hence if a[`(b)]\alpha \bar {\beta } is in H^￥ + CH^\infty + C then ||S_a, b ||_e = max(||a||_￥ , ||b||_￥ ).\Vert S_{\alpha ,\,\beta } \Vert _e = \max (\Vert \alpha \Vert _\infty , \Vert \beta \Vert _\infty ). This gives a known result when a, b\alpha , \beta are in C.     

Global convergence and the Powell singular function

The Powell singular function was introduced 1962 by M.J.D. Powell as an unconstrained optimization problem. The function is also used as nonlinear least squares problem and system of nonlinear equations. The function is a classic test function included in collections of test problems in optimization as well as an example problem in text books. In the global optimization literature the function is stated as a difficult test case. The function is convex and the Hessian has a double singularity at the solution. In this paper we consider Newton’s method and methods in Halley class and we discuss the relationship between these methods on the Powell Singular Function. We show that these methods have global but linear rate of convergence. The function is in a subclass of unary functions and results for Newton’s method and methods in the Halley class can be extended to this class. Newton’s method is often made globally convergent by introducing a line search. We show that a full Newton step will satisfy many of standard step length rules and that exact line searches will yield slightly faster linear rate of convergence than Newton’s method. We illustrate some of these properties with numerical experiments.     

On the error estimate for calculating some singular integrals

We consider some classes of multi-dimensional singular integral operators with Calderon-Zygmund kernels, construct their discrete analogues and give the error estimate for approximation of continual singular integral operator by the discrete ones. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Multi-peak solutions for some singular perturbation problems

We consider the problem where is a smooth domain in , not necessarily bounded, is a small parameter and f is a superlinear, subcritical nonlinearity. It is known that this equation possesses a solution that concentrates, as approaches zero, at a maximum of the function , the distance to the boundary. We obtain multi-peak solutions of the equation given above when the domain presents a distance function to its boundary d with multiple local maxima. We find solutions exhibiting concentration at any prescribed finite set of local maxima, possibly degenerate, of d. The proof relies on variational arguments, where a penalization-type method is used together with sharp estimates of the critical values of the appropriate functional. Our main theorem extends earlier results, including the single peak case. We allow a degenerate distance function and a more general nonlinearity. Received September 3, 1998 / Accepted February 29, 1999     

On some singular perturbation problems in the theory of lubrication

Some singular perturbation problems occurring in the theory of hydrodynamic lubrication are studied. We examine in detail the case of the very long or very short journal bearing; the asymptotic problem of small eccentricity ratio is also considered.     

The Calkin image of algebras of singular integral operators

The image in the Calkin algebra of the Banach algebra generated by all singular integral operators with piecewise continuous coefficients on some composed contour is described up to isomorphy and isometry for weighted L^P-spaces.     

Chromatic sums of singular maps on some surfaces

A map is singular if each edge is on the same face on a sruface (i.e., those have only one face on a surface). Because any map with loop is not colorable, all maps here are assumed to be loopless. In this paper povides the explicit expression of chromatic sum functions for rooted singular maps on the projective plane, the torus and the Klein bottle. From the explicit expression of chromatic sum functions of such maps, the explicit expression of enum erating functions of such maps are also derived.     

Applications of some block spaces to singular integrals

This paper is a survey on the theory and application of some block spaces on the unit sphere introduced by Jiang and the author of this paper in the study of singular integrals and some related operators with rough kernels.