Analysis of a dynamic viscoelastic unilateral contact problem with normal damped response

In this paper we deal with a viscoelastic unilateral contact problem with normal damped response. The process is assumed to be dynamic and frictionless. Normal damping function is modeled by the Clarke subdifferential of a nonconvex and nonsmooth function. First, the variational formulation of this problem is provided in the form of a nonlinear first order variational–hemivariational inequality for the velocity field. Then, based on the surjectivity results for pseudomonotone and maximal monotone operators, we obtain the unique solvability for a new class of abstract evolutionary variational-hemivariational inequalities. Finally, we apply our abstract results to prove the existence of a unique weak solution to the corresponding contact problem.     

Local solvability of linear differential operators with double characteristics. I. Necessary conditions

This is a the first in a series of two articles devoted to the question of local solvability of doubly characteristic differential operators L, defined, say, in an open set $\Omega \subset \mathbb{R}^n.This is a the first in a series of two articles devoted to the question of local solvability of doubly characteristic differential operators L, defined, say, in an open set Suppose the principal symbol p _k of L vanishes to second order at , and denote by the Hessian form associated to p _k on . As the main result of this paper, we show (under some rank conditions and some mild additional conditions) that a necessary condition for local solvability of L at x ₀ is the existence of some such that . We apply this result in particular to operators of the formwhere the X _j are smooth real vector fields and the α _jk are smooth complex coefficients forming a symmetric matrix . We say that L is essentially dissipative at x ₀, if there is some such that e ^iθ L is dissipative at x ₀, in the sense that . For a large class of doubly characteristic operators L of this form, our main result implies that a necessary condition for local solvability at x ₀ is essential dissipativity of L at x ₀. By means of H?rmander’s classical necessary condition for local solvability, the proof of the main result can be reduced to the following question: suppose that Q _A and Q _B are two real quadratic forms on a finite dimensional symplectic vector space, and let Q _C : = {Q _A ,Q _B } be given by the Poisson bracket of Q _A and Q _B . Then Q _C is again a quadratic form, and we may ask: when can we find a common zero of Q _A and Q _B at which Q _C does not vanish? The study of this question occupies most of the paper, and the answers may be of independent interest. In the second paper of this series, building on joint work with F. Ricci, M. Peloso and others, we shall study local solvability of essentially dissipative left-invariant operators of the form (0.1) on Heisenberg groups in a fairly comprehensive way. Various examples exhibiting a kind of exceptional behaviour from previous joint works, e.g., with G. Karadzhov, have shown that there is little hope for a complete characterization of locally solvable operators on Heisenberg groups. However, the “generic” scheme of what rules local solvability of second order operators on Heisenberg groups becomes evident from our work.      

Solvability and optimal controls for semilinear fractional evolution hemivariational inequalities

This paper is concerned with the control systems of semilinear fractional evolution hemivariational inequalities and their optimal controls in Banach space. Firstly, the existence of mild solutions is obtained and proved mainly by using a well‐known fixed point theorem of multivalued maps and the properties of generalized Clarke subdifferential. Then, by applying generally mild conditions of cost functionals, we investigate the existence results of the optimal controls for fractional differential evolution hemivariational inequalities. Finally, an example is given to demonstrate the applicability of the main results. Copyright © 2016 John Wiley & Sons, Ltd.     

On the n-minimal subgroups

Let G be a finite group and n be a positive integer. An n-minimal subgroup H of G is called to be exactly n-minimal if no proper subgroup of H is n-minimal. In this paper, we study the solvability of G under the assumption that all exactly n-minimal subgroups of G are S-permutable.     

A mixed‐type Galerkin variational formulation and fast algorithms for variable‐coefficient fractional diffusion equations

We consider the variable‐coefficient fractional diffusion equations with two‐sided fractional derivative. By introducing an intermediate variable, we propose a mixed‐type Galerkin variational formulation and prove the existence and uniqueness of the variational solution over . On the basis of the formulation, we develop a mixed‐type finite element procedure on commonly used finite element spaces and derive the solvability of the finite element solution and the error bounds for the unknown and the intermediate variable. For the Toeplitz‐like linear system generated by discretization, we design a fast conjugate gradient normal residual method to reduce the storage from O(N²) to O(N) and the computing cost from O(N³) to O(NlogN). Numerical experiments are included to verify our theoretical findings. Copyright © 2017 John Wiley & Sons, Ltd.     

Global solvability of the Cauchy characteristic problem for one class of nonlinear second order hyperbolic systems

The Cauchy characteristic problem in the light cone of the future for one class of nonlinear hyperbolic systems of the second order is considered. The existence and uniqueness of global solution of this problem is proved.     

On the Microlocal Properties of the Range of Systems of Principal Type

The purpose of this paper is to study microlocal conditions for inclusion relations between the ranges of square systems of pseudodifferential operators which fail to be locally solvable. The work is an extension of earlier results for the scalar case in this direction, where analogues of results by L. Hörmander about inclusion relations between the ranges of first order differential operators with coefficients in C ^∞ which fail to be locally solvable were obtained. We shall study the properties of the range of systems of principal type with constant characteristics for which condition (Ψ) is known to be equivalent to microlocal solvability.