Estimates for solutions of Burgers type equations and some applications

We obtain precise large time asymptotics for the Cauchy problem for Burgers type equations satisfying shock profile condition. The proofs are based on the exact a priori estimates for (local) solutions of these equations and the result of [G.M. Henkin, A.A. Shananin, Asymptotic behavior of solutions of the Cauchy problem for Burgers type equations, J. Math. Pures Appl. 83 (2004) 1457–1500].     

Estimates for imaginary powers of Laplace operator in variable Lebesgue spaces and applications

In this paper we study some estimates of norms in variable exponent Lebesgue spaces for singular integral operators that are imaginary powers of the Laplace operator in ? ⁿ . Using the Mellin transform argument, fromthese estimates we obtain the boundedness for a family of maximal operators in variable exponent Lebesgue spaces, which are closely related to the (weak) solution of the wave equation.     

Estimates for the Green function of a general Sturm-Liouville operator and their applications

For a general Sturm-Liouville operator with nonnegative coefficients, we obtain two-sided estimates for the Green function, sharp by order on the diagonal.

     

A control operator and some of its applications

The conventional function space algorithms for solving minimization of penalized cost functional for optimal control problem characterized by linear-system integral quadratic cost due to Di Pillo and others, though falling within the framework pertinent to the conjugate gradient method algorithm, is difficult to apply computationally. The difficulty arises principally because there exists in the algorithm a number of stringent requirements imposed on the minimization procedures to facilitate its convergence. Incidentally, such computations are very cumbersome to carry out numerically. To circumvent this major numerical draw back, we construct here a control operator associated with this class of problems and use our explicit knowledge of the operator to devise an extended conjugate gradient method algorithm for solving this family of problems. Furthermore, the establishment of some functional inequalities which are obtained using the knowledge of the control operator is discussed.     

Estimates for norms of matrix-valued and operator-valued functions and some of their applications

A survey is presented of estimates for a norm of matrix-valued and operator-valued functions obtained by the author. These estimates improve the Gel'fand-Shilov estimate for regular functions of matrices and Carleman's estimates for resolvents of matrices and compact operators.From the estimates for resolvents, the well-known result for spectrum perturbations of self-adjoint operators is extended to quasi-Hermitian operators. In addition, the classical Schur and Brown's inequalities for eigenvalues of matrices are improved.From estimates for the exponential function (semigroups), bounds for solution norms of nonlinear differential equations are derived. These bounds give the stability criteria which make it possible to avoid the construction of Lyapunov functions in appropriate situations.     

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.     

Estimates for eigenvalues of the Paneitz operator

For an n  -dimensional compact submanifold Mⁿ$M^n$ in the Euclidean space R^N${\mathbf{R}}^N$, we study estimates for eigenvalues of the Paneitz operator on Mⁿ$M^n$. Our estimates for eigenvalues are sharp.     

Estimates for the Hill operator, II

Consider the Hill operator Ty=-y^″+q(t)y in L²(R), where the real potential q is 1-periodic and q,q^′∈L²(0,1). The spectrum of T consists of spectral bands separated by gaps γ_n,n?1 with length |γ_n|?0. We obtain two-sided estimates of the gap lengths ∑n²|γ_n|² in terms of . Moreover, we obtain the similar two-sided estimates for spectral data (the height of the corresponding slit on the quasimomentum domain, action variables for the KdV equation and so on). In order prove this result we use the analysis of a conformal mapping corresponding to quasimomentum of the Hill operator. That makes it possible to reformulate the problems for the differential operator as the problems of the conformal mapping theory. Then the proof is based on the analysis of the conformal mapping, the embedding theorems and the identities. Furthermore, we obtain the similar two-sided estimates for potentials which have p?2 derivatives.     

Estimates for eigenvalues of fourth-order weighted polynomial operator

In this paper,we investigate the Dirichlet eigenvalue problem of fourth-order weighted polynomial operator △~2u-a△u+bu=Λρu,inΩR~n,u|Ω=uvΩ=0,where the constants a,b≥0.We obtain some estimates for the upper bounds of the (k+1)-th eigenvalueΛ_k+1 in terms of the first k eigenvalues.Moreover,these results contain some results for the biharmonic operator.     

Estimates on some power nonlinear Volterra-Fredholm type discrete inequalities and their applications

Some explicit bounds on solutions to a class of new power nonlinear Volterra-Fredholm type discrete inequalities are established, which can be used as effective tools in the study of certain sum-difference equations. Application examples are also given.     

Estimates for the Levy-Prokhorov distance in terms of characteristic functions and some of their applications

One gives estimates of the Levy-Prokhorov distance between distributions in R^k in terms of integrals of the differences of the characteristic functions and of their derivatives, similar to Esseen's inequality.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 119, pp. 108–127, 1982.The author is grateful to I. A. Ibragimov for his interest in this paper.     

Inertia theorems for operator pencils and applications

In this paper we obtain general infinite dimensional inertia theorems for linear pencils in Hilbert space which cover previously known results for the finite dimensional case and for block weighted shifts. Connections with definite subspaces for contractions in spaces with indefinite metric are discussed.     

Estimates for eigenvalues of Laplacian operator with any order

Let D be a bounded domain in an n-dimensional Euclidean space Rn. Assume that 0 ＜ λ1 ≤λ2 ≤ … ≤ λκ ≤ … are the eigenvalues of the Dirichlet Laplacian operator with any order l{(-△)lu=λu, in D u=(δ)u/(δ)(→n)=…(δ)l-1u/(δ)(→n)l-1=0,on (δ)D.Then we obtain an upper bound of the (k 1)-th eigenvalue λκ 1 in terms of the first k eigenvalues.k∑i=1(λκ 1-λi) ≤ 1/n[4l(n 2l-2)]1/2{k∑i=1(λκ 1-λi)1/2λil-1/l k∑i=1(λκ 1-λi)1/2λ1/li}1/2.This ineguality is independent of the domain D. Furthermore, for any l ≥ 3 the above inequality is better than all the known results. Our rusults are the natural generalization of inequalities corresponding to the case l = 2 considered by Qing-Ming Cheng and Hong-Cang Yang. When l = 1, our inequalities imply a weaker form of Yang inequalities. We aslo reprove an implication claimed by Cheng and Yang.     

Estimates on the solution of a schrödinger equation and some applications

Dedicated to Elliott Lieb on the occasion of his 60th birthday.     

Estimates of operator moduli of continuity

In Aleksandrov and Peller (2010) [2] we obtained general estimates of the operator moduli of continuity of functions on the real line. In this paper we improve the estimates obtained in Aleksandrov and Peller (2010) [2] for certain special classes of functions. In particular, we improve estimates of Kato (1973) [18] and show that

     

Estimates for some convolution operators with singularities in their kernels on a sphere with applications

We study convolution operators whose kernels have singularities on the unit sphere. For these operators we obtainH ^p -H ^q estimates, where p is less or equal q, and prove their sharpness. To this end, we develop a new method that uses special representations for the symbol of such operators as sums of certain oscillatory integrals and applies the stationary phase method and A. Miyachi results for model oscillating multipliers. Moreover, we also obtain estimates for operators from L ^p to BMO and those from BMO to BMO.