Completeness Theorems on the Boundary for a Parabolic Equation

Let α\ { v α } be a system of polynomial solutions of the parabolic equation aₕₖ∂ _xₕxₖu - ∂ ₜ u =0 a hk ∂ x h x k u - ∂ t u = 0 in a bounded C¹ C 1 -cylinder Ω T Ω T contained in Rⁿ⁺¹ R n + 1 . Here, aₕₖ∂ _xₕxₖ a hk ∂ x h x k is an elliptic operator with real constant coefficients. We prove that α\ { v α } is complete in Lᵖ(Σ ') L p ( Σ ′ ) , where Σ ' Σ ′ is the parabolic boundary of Ω T Ω T . Similar results are proved for the adjoint equation aₕₖ∂ _xₕxₖ u+ ∂ ₜ u =0 a hk ∂ x h x k