Some results on eigenvalue problems in the theory of piezoelectric porous dipolar bodies

In our study we construct a boundary value problem in elasticity of porous piezoelectric bodies with a dipolar structure To construct an eigenvalue problem in this context, we consider two operators defined on adequate Hilbert spaces. We prove that the two operators are positive and self adjoint, which allowed us to show that any eigenvalue is a real number and two eigenfunctions which correspond to two distinct eigenvalues are orthogonal. With the help of a Rayleigh quotient type functional, a variational formulation for the eigenvalue problem is given. Finally, we consider a disturbation analysis in a particular case. It must be emphasized that the porous piezoelectric bodies with dipolar structure addressed in this study are considered in their general form, i.e.,inhomogeneous and anisotropic.

Automated summary

In this study, we establish a boundary value problem for elasticity of porous piezoelectric bodies with a dipolar structure. By defining two operators on suitable Hilbert spaces, we prove their positivity and self-adjointness, demonstrating that eigenvalues are real numbers and corresponding eigenfunctions are orthogonal. A variational formulation is provided for the eigenvalue problem using a Rayleigh quotient type functional. Additionally, a disturbance analysis in a specific case is investigated. It is important to note that the porous piezoelectric bodies with dipolar structure considered in this study are of a general form, i.e., inhomogeneous and anisotropic.