On the boundary operator of a fourth order differential problem

We study an unilateral elliptic fourth order problem and establish some uniqueness results. The expression of the boundary operation is also given     

Homotopy perturbation method for modified Camassa-Holm and Degasperis-Procesi equations

In this Letter, homotopy perturbation method (HPM) is directly applied to modified Camassa-Holm and Degasperis-Procesi equations to obtain their solitary wave solutions. The results show the applicability, accuracy and efficiency of HPM in solving nonlinear differential equations with fully nonlinear dispersion term. It is predicted that HPM can be widely applied in engineering problems.     

Short time existence and uniqueness in Hölder spaces for the 2D dynamics of dislocation densities

In this paper, we study the model of Groma and Balogh [I. Groma, P. Balogh, Investigation of dislocation pattern formation in a two-dimensional self-consistent field approximation, Acta Mater. 47 (1999) 3647–3654] describing the dynamics of dislocation densities. This is a two-dimensional model where the dislocation densities satisfy a system of two transport equations. The velocity vector field is the shear stress in the material solving the equations of elasticity. This shear stress can be related to Riesz transforms of the dislocation densities. Basing on some commutator estimates type, we show that this model has a unique local-in-time solution corresponding to any initial datum in the space Cr(R²)∩Lp(R²)$C^r({\mathbb{R}}^2)\cap L^p({\mathbb{R}}^2)$ for r>1$r>1$ and 1<p<+∞$1<p<+\infty$, where Cr(R²)$C^r({\mathbb{R}}^2)$ is the Hölder–Zygmund space.     

Extended Polar Decompositions for Plane Strain

In a finite deformation at a particle of a continuous body, a triad of infinitesimal material line elements is said to be “unsheared” when the angles between the three pairs of line elements of the triad suffer no change. In a previous paper, it has been shown that there is an infinity of unsheared oblique triads. With each oblique unsheared triad may be associated an “extended polar decomposition” F = QG = HQ of the deformation gradient F, in which Q is a rotation tensor, and G, H are not symmetric. Both G and H have the same real eigenvalues which are the stretches of the elements of the triad. In this paper, a detailed analysis of extended polar decompositions is presented in the case when the finite deformation is that of plane strain. Then, we may deal with a 2 × 2 deformation gradient F′ = Q′G′ = H′Q′ instead of the full 3 × 3 tensor F. In this case, the extended polar decompositions are associated with “unsheared pairs,” i.e., pairs of infinitesimal material line elements in the plane of strain which suffer no change in angle in the deformation. If one arm of an unsheared pair is chosen in the plane of strain, then, in general, its companion in the plane is determined. It follows that all possible extended polar decompositions may then be described in terms of a single parameter, the angle that the chosen arm makes with a coordinate axis in the plane. Explicit expressions for G′ and H′ are obtained, and various special cases are discussed. In particular, we note that the expressions for G′ and H′ remain valid even when the chosen arm is along a “limiting direction,” that is the direction of a line element which has no companion element in the plane forming an unsheared pair with it. The results are illustrated by considering the cases of simple shear and of pure shear.Dedicated to Professor Piero Villaggio as a symbol of our friendship and esteem.     

On the Canonical Elastic Moduli of Linear Plane Anisotropic Elasticity

The linear, planar, anisotropic elastic equilibrium equations are transformed to canonical form, through linear transformations of both coordinates and unknown displacement functions, together with a linear combination of equations. Correspondingly, the six original material moduli are replaced by two canonical elastic moduli. Similar results have been reached by Olver in 1988. However, the method demonstrated in this paper is more concise and direct. As an example, the general solution to the canonical equations is obtained in the case of a pair of double roots.