Analytic Solutions and Solvability of the Polyharmonic Cauchy Problem in Rn

This study develops a rigorous analytic framework for solving the Cauchy problem of
polyharmonic equations in Rn, highlighting the crucial role of symmetry in the structure,
stability, and solvability of solutions. Polyharmonic equations, as higher-order extensions
of Laplace and biharmonic equations, frequently arise in elasticity, potential theory,
and mathematical physics, yet their Cauchy problems are inherently ill-posed. Using
hyperspherical harmonics and homogeneous harmonic polynomials, whose orthogonality
reflects the underlying rotational and reflectional symmetries, the study constructs explicit,
uniformly convergent series solutions. Through analytic continuation of integral representations,
necessary and sufficient solvability criteria are established, ensuring convergence
of all derivatives on compact domains. Furthermore, newly derived Green-type identities
provide a systematic method to reconstruct boundary information and enforce stability
constraints. This approach not only generalizes classical Laplace and biharmonic results
to higher-order polyharmonic equations but also demonstrates how symmetry governs
boundary data admissibility, convergence, and analytic structure, offering both theoretical
insights and practical tools for elasticity, inverse problems, and mathematical physics.