Abstract
We consider an inverse scattering problem of recovering the unknown
coefficients of a quasi-linearly perturbed biharmonic operator in the three-dimensional case.
These unknown complex-valued coefficients are assumed to satisfy some regularity
conditions on their nonlinearity, but they can be discontinuous or singular
in their space variable. We prove Saito’s formula and uniqueness theorem
of recovering some essential information about the unknown coefficients
from the knowledge of the high frequency scattering amplitude.
The spatial properties of solutions for a class of thermoelastic plate with biharmonic operator were studied. The energy method was used. We constructed an energy expression. A differential inequality which the energy expression was controlled by a second-order differential inequality is deduced. The Phragme´n-Lindelo¨f alternative results of the solutions were obtained by solving the inequality. These results show that the Saint-Venant principle is also valid for the hyperbolic–hyperbolic coupling equations. Our results can been seen as a version of symmetry in inequality for studying the Phragme´n-Lindelo¨f alternative results.
Abstract
Differential equations with variable exponent arise from the nonlinear elasticity theory and the theory of electrorheological fluids.
We study the existence of at least three weak solutions for the nonlocal elliptic problems driven by
p
(
x
)
p(x)
-biharmonic operator.
Our technical approach is based on variational methods.
Some applications illustrate the obtained results.
We also provide an example in order to illustrate our main abstract results.
We extend and improve some recent results.
AbstractThis article offers a study of the Calderón type inverse problem of determining up to second order coefficients of higher order elliptic operators. Here we show that it is possible to determine an anisotropic second order perturbation given by a symmetric matrix, along with a first order perturbation given by a vector field and a zero-th order potential function inside a bounded domain, by measuring the Dirichlet to Neumann map of the perturbed biharmonic operator on the boundary of that domain.
A nonlocal analogue of the biharmonic operator with involution-type transformations was considered. For the corresponding biharmonic equation with involution, we investigated the solvability of boundary value problems with a fractional-order boundary operator having a derivative of the Hadamard-type. First, transformations of the involution type were considered. The properties of the matrices of these transformations were investigated. As applications of the considered transformations, the questions about the solvability of a boundary value problem for a nonlocal biharmonic equation were studied. Modified Hadamard derivatives were considered as the boundary operator. The considered problems covered the Dirichlet and Neumann-type boundary conditions. Theorems on the existence and uniqueness of solutions to the studied problems were proven.
In this paper, using the variational principle, the existence and multiplicity of solutions for
p
x
,
q
x
-Kirchhoff type problem with Navier boundary conditions are proved. At the same time, the sufficient conditions for the multiplicity of solutions are obtained.