Nonlinear quasistatic problems of gradient type in inelastic deformations theory

In this paper we study nonlinear quasistatic problems from inelastic deformations theory. Only strictly monotone, gradient-type constitutive equations are considered. We prove existence for both coercive and non-coercive models, using energy estimates and Young measures. For non-coercive models we use the L² self-controlling property.     

KIRCHHOFF TYPE EQUATIONS DEPENDING ON A SMALL PARAMETER

ＫＩＲＣＨＨＯＦＦＴＹＰＥＥＱＵＡＴＩＯＮＳＤＥＰＥＮＤＩＮＧＯＮＡＳＭＡＬＬＰＡＲＡＭＥＴＥＲ￥Ｐ．Ｄ＇ＡＮＣＯＮＡＳ．ＳＰＡＧＮＯＬＯ（ＤｅｐａｒｔｍｅｌｌｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＰｉｓａＵｎｉｖｅｒｓｉｔｙ！Ｐｉｓａ，Ｉｔａｌｙ）Ａｂｓｔ...     

Identification of small amplitude perturbations in the electromagnetic parameters from partial dynamic boundary measurements

We consider the inverse problem of reconstructing small amplitude perturbations in the conductivity for the wave equation from partial (on part of the boundary) dynamic boundary measurements. Through construction of appropriate test functions by a geometrical control method we provide a rigorous derivation of the inverse Fourier transform of the perturbations in the conductivity as the leading order of an appropriate averaging of the partial dynamic boundary perturbations. This asymptotic formula is generalized to the full time-dependent Maxwell's equations. Our formulae may be expected to lead to very effective computational identification algorithms, aimed at determining electromagnetic parameters of an object based on partial dynamic boundary measurements.     

Mathematical analysis of successive linear approximation for Mooney-Rivlin material model in finite elasticity

For calculating large deformations of solid bodies, we have proposed a method of successive linear approximation, by considering the relative Lagrangian formulation. In this article we briefly describe this method, which is applied for nearly incompressible Mooney-Rivlin materials. We prove the existence and uniqueness of weak solutions for associated boundary value problems that arise in each step of the method. In our analysis we consider also a non-standard case, where the coefficients present in the constitutive function of Mooney-Rivlin materials do not satisfy the usual E-inequalities.     

Infinitely many solutions for nonlinear Schrödinger equations with electromagnetic fields

In this paper, we study the nonlinear Schrödinger equation with electromagnetic fields

     

Positive solutions of a nonlocal singular elliptic equation by means of a non-standard bifurcation theory

We consider a nonlocal elliptic equation arising in a prey–predator model whose nonlocal term is singular. We use the Leray–Schauder degree to prove the existence of an unbounded continuum of positive solutions emanating from the trivial solution. As application, we study nonlocal and singular elliptic equations of the type logistic and Holling–Tanner.     

Three dimensional electromagnetic scattering T-matrix computations

The infinite T-matrix method is a powerful tool for electromagnetic scattering simulations, particularly when one is interested in changes in orientation of the scatterer with respect to the incident wave or changes of configuration of multiple scatterers and random particles, because it avoids the need to solve the fully reconfigured systems. The truncated T-matrix (for each scatterer in an ensemble) is often computed using the null-field method. The main disadvantage of the null-field based T-matrix computation is its numerical instability for particles that deviate from a sphere. For large and/or highly non-spherical particles, the null-field method based truncated T-matrix computations can become slowly convergent or even divergent. In this work, we describe an electromagnetic scattering surface integral formulation for T-matrix computations that avoids the numerical instability. The new method is based on a recently developed high-order surface integral equation algorithm for far field computations using basis functions that are tangential on a chosen non-spherical obstacle. The main focus of this work is on the mathematical details required to apply the high-order algorithm to compute a truncated T-matrix that describes the scattering properties of a chosen perfect conductor in a homogeneous medium. We numerically demonstrate the stability and convergence of the T-matrix computations for various perfect conductors using plane wave incident radiation at several low to medium frequencies and simulation of the associated radar cross of the obstacles.     

On a class of parabolic equations with nonlocal boundary conditions

In this paper we study a class of parabolic equations subject to a nonlocal boundary condition. The problem is a generalized model for a theory of ion-diffusion in channels. By using energy method, we first derive some a priori estimates for solutions and then prove that the problem has a unique global solution. Moreover, under some assumptions on the nonlinear boundary condition, it is shown that the solution blows up in finite time. Finally, the long-time behavior of solution to a linear problem is also studied in the paper.     

A sufficient condition for the existence of a principal eigenvalue for nonlocal diffusion equations with applications

Considerable work has gone into studying the properties of nonlocal diffusion equations. The existence of a principal eigenvalue has been a significant portion of this work. While there are good results for the existence of a principal eigenvalue equations on a bounded domain, few results exist for unbounded domains. On bounded domains, the Krein–Rutman theorem on Banach spaces is a common tool for showing existence. This article shows that generalized Krein–Rutman can be used on unbounded domains and that the theory of positive operators can serve as a powerful tool in the analysis of nonlocal diffusion equations. In particular, a useful sufficient condition for the existence of a principal eigenvalue is given.     

Universal Models of Soliton Hierarchies

We consider commutativity equations and a related new model similar to the Kadomtsev–Petviashvili equation in the Sato theory. Integration of this model equation with three independent variables is based on a generalization of the Dubrovin equations and the recently developed theory of transformations of spectral problems. We give examples of equations with a fractional-power dispersion law that can be linearized in this theory.     

Existence of positive solution for a system of elliptic equations via bifurcation theory

In this paper we study the existence of solution for the following class of elliptic systems

(P)$\{\begin{array}{c}?\mathrm{\Delta }u=(a?{\int }_{\mathrm{\Omega }}K(x,y)f(u,v)dy)u+bv,\text{in}\mathrm{\Omega }\hfill \\ ?\mathrm{\Delta }v=(d?{\int }_{\mathrm{\Omega }}\mathrm{\Gamma }(x,y)g(u,v)dy)v+cu,\text{in}\mathrm{\Omega }\hfill \\ u=v=0,\text{on}?\mathrm{\Omega }\hfill \end{array}$

where $\mathrm{\Omega }?{\mathbb{R}}^N$ is a smooth bounded domain, $N\ge 1$, and $K,\mathrm{\Gamma }:\mathrm{\Omega }\times \mathrm{\Omega }\to \mathbb{R}$ are nonnegative functions satisfying some hypotheses and $a,b,c,d\in \mathbb{R}$. The functions f and g satisfy some conditions which permit to use Bifurcation Theory to prove the existence of solution for (P).     

Nonlocal symmetries and recursion operators: Partial differential and differential-difference equations

A systematic method to derive the nonlocal symmetries for partial differential and differential-difference equations with two independent variables is presented and shown that the Korteweg-de Vries (KdV) and Burger's equations, Volterra and relativistic Toda (RT) lattice equations admit a sequence of nonlocal symmetries. An algorithm, exploiting the obtained nonlocal symmetries, is proposed to derive recursion operators involving nonlocal variables and illustrated it for the KdV and Burger's equations, Volterra and RT lattice equations and shown that the former three equations admit factorisable recursion operators while the RT lattice equation possesses (2×2) matrix factorisable recursion operator. The existence of nonlocal symmetries and the corresponding recursion operator of partial differential and differential-difference equations does not always determine their mathematical structures, for example, bi-Hamiltonian representation.     

C 1, α Interior Regularity for Nonlinear Nonlocal Elliptic Equations with Rough Kernels

We prove a C ^{1, α} interior regularity theorem for fully nonlinear uniformly elliptic integro-differential equations without assuming any regularity of the kernel. We then give some applications to linear theory and higher regularity of a special class of nonlinear operators.     

Harnack's inequality for parabolic nonlocal equations

The main result of this paper is a nonlocal version of Harnack's inequality for a class of parabolic nonlocal equations. We additionally establish a weak Harnack inequality as well as local boundedness of solutions. None of the results require the solution to be globally positive.     

Complete symmetry group and nonlocal symmetries for some two-dimensional evolution equations

The complete symmetry group of an 1+1 evolution equation of maximal symmetry has been demonstrated to be represented by the six-dimensional Lie algebra of point symmetries sl(2,R)s_⊕W, where W is the three-dimensional Heisenberg-Weyl algebra. We construct a complete symmetry group of a 1+2 evolution equation ut=(Fy(u)ux) for some functions F using the point symmetries admitted by the equation. The 1+2 equation is not completely specifiable by point symmetries alone for some specific functions F. We make use of Ansätze already reported by Myeni and Leach [S.M. Myeni, P.G.L. Leach, Nonlocal symmetries and complete symmetry groups of evolution equations, J. Nonlinear Math. Phys. 13 (2006) 377-392] which provide a route to the determination of the required generic nonlocal symmetries necessary to supplement the point symmetries for the complete specification of these 1+2 evolution equations. Further we find that taking some suitable linear combination of Lie point symmetries helps to optimise the procedure of specifying the equation. A general result concerning the number of symmetries required to form a complete symmetry group of evolution is presented in the Conclusion.     

ON SOME MODEL DIFFUSION PROBLEMS WITH A NONLOCAL LOWER ORDER TERM

The authors consider a class of nonlinear parabolic problems where the lower order term is depending on a weighted integral of the solution, and address the issues of existence, uniqueness, stationary solutions and in some cases asymptotic behaviour.