The problem of finding eigenvalues and eigenfunctions of boundary value problems for an equation of mixed type
Abstract
Abstract In this work, we investigate the eigenvalue problem for an elliptic-hyperbolic equation with a non-smooth line of type changing within a special domain. The boundary conditions considered include Frankl’s condition, which is crucial for such problems involving mixed type equations. We derive the eigenvalues and corresponding eigenfunctions of the problem, paying special attention to the structural complexities introduced by the discontinuity in the equation’s type across the domain. Furthermore, we rigorously prove that the system of eigenfunctions obtained from the problem is not complete in the L 2 space, meaning that not every square-integrable function in the domain can be represented as a series expansion in terms of these eigenfunctions. This non-completeness is demonstrated by constructing a specific function orthogonal to the entire system of eigenfunctions. We utilize advanced techniques, including the transformation to polar coordinates and variable substitution, to express integrals involving Bessel functions, thereby providing a clear framework for proving the non-completeness. Our findings are applicable to both symmetric and anti-symmetric cases of the problem, with boundary conditions that either involve equal or opposite values at points reflected across a line of symmetry. This study highlights significant challenges in dealing with mixed-type equations and contributes to the broader understanding of how eigenvalue problems with Frankl’s conditions behave in non-standard geometries, particularly in domains with discontinuities.