scholarly journals Elliptic equations with absorption in a half-space

2016 ◽  
Author(s):  
J. García-Melián ◽  
A. Quaas ◽  
B. Sirakov
Keyword(s):  
Half Space ◽  
Elliptic Equations

A long-wave model for the surface elastic wave in a coated half-space

2010 ◽  
Vol 466 (2122) ◽  
pp. 3097-3116 ◽  
Author(s):  
H.-H. Dai ◽  
J. Kaplunov ◽  
D. A. Prikazchikov
Keyword(s):  
Surface Wave ◽  
Boundary Conditions ◽  
Half Space ◽  
Elliptic Equations ◽  
Ad Hoc ◽  
Perturbation Technique ◽  
Three Dimensional ◽  
Wave Model ◽  
Destructive Testing ◽  
Long Wave

The paper deals with the three-dimensional problem in linear isotropic elasticity for a coated half-space. The coating is modelled via the effective boundary conditions on the surface of the substrate initially established on the basis of an ad hoc approach and justified in the paper at a long-wave limit. An explicit model is derived for the surface wave using the perturbation technique, along with the theory of harmonic functions and Radon transform. The model consists of three-dimensional ‘quasi-static’ elliptic equations over the interior subject to the boundary conditions on the surface which involve relations expressing wave potentials through each other as well as a two-dimensional hyperbolic equation singularly perturbed by a pseudo-differential (or integro-differential) operator. The latter equation governs dispersive surface wave propagation, whereas the elliptic equations describe spatial decay of displacements and stresses. As an illustration, the dynamic response is calculated for impulse and moving surface loads. The explicit analytical solutions obtained for these cases may be used for the non-destructive testing of the thickness of the coating and the elastic moduli of the substrate.

scholarly journals Another approach of Morrey estimate for linear elliptic equations with partially BMO coefficients in a half space

Filomat ◽  
2018 ◽  
Vol 32 (4) ◽  
pp. 1429-1437
Author(s):  
Hong Tian ◽  
Shenzhou Zheng
Keyword(s):  
Half Space ◽  
Elliptic Equations ◽  
Elementary Approach ◽  
Dirichlet Problems ◽  
Spatial Variables ◽  
Morrey Estimates ◽  
Lp Estimate ◽  
Special Weight ◽  
Bmo Coefficients ◽  
Weak Derivatives

Making use of an elementary approach instead of the weighted Lp estimate with a special weight, we prove global Morrey estimates of the weak derivatives to the Dirichlet problems of linear elliptic equations with small partially BMO coefficients in a half space. Here, the leading coefficients aij(x) are assumed to be merely measurable in one variable, and have small BMO in the remaining spatial variables.

Asymptotic behaviour and symmetry of positive solutions to nonlinear elliptic equations in a half-space

2016 ◽  
Vol 146 (6) ◽  
pp. 1243-1263 ◽  
Author(s):  
Lei Wei
Keyword(s):  
Half Space ◽  
Positive Solutions ◽  
Elliptic Equations ◽  
Nonlinear Elliptic Equations ◽  
Unique Positive Solution ◽  
Bounded Smooth Domain ◽  
Nonlinear Elliptic ◽  
Bounded Smooth ◽  
Image Position ◽  
General Nonlinearity

We consider the following equation:where d(x) = d(x, ∂Ω), θ > –2 and Ω is a half-space. The existence and non-existence of several kinds of positive solutions to this equation when , f(u) = up(p > 1) and Ω is a bounded smooth domain were studied by Bandle, Moroz and Reichel in 2008. Here, we study exact the behaviour of positive solutions to this equation as d(x) → 0+ and d(x) → ∞, respectively, and the symmetry of positive solutions when , Ω is a half-space and f(u) is a more general nonlinearity term than up. Under suitable conditions for f, we show that the equation has a unique positive solution W, which is a function of x1 only, and W satisfies

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