ABSTRACT
As in 1D geometry, we begin with a class of quadratic second-order operators admitting polynomial subspaces. Consider a family of operators given by
F[u] = α( u)2 + βu u + γ |∇u|2 in IRN , (6.1) where, as usual, α, β, and γ are arbitrary parameters. Here x = (x1, x2, ..., xN )T is a vector in IRN . Obviously, F preserves the subspace of linear functions
W linN = L{x1, ..., xN }. (6.2) This defines a simple map F : W linN → L{1}. The next observation is also easy. Proposition 6.1 Operator (6.1) preserves: (i) the 2D subspace of radial functions
W r2 = L{1, |x |2}; (6.3) (ii) the (N+1)-dimensional subspace of diagonal quadratic forms
W qN+1 = L { 1, x21 , ..., x
2 N }; (6.4)
(iii) the subspace of arbitrary quadratic forms
W qn = L{1, xi x j , 1 ≤ i, j ≤ N} ( n = 1+ N(N+1)2
); and (6.5) (vi) the direct sum of subspaces (6.2) and (6.5),
W qn ⊕ W linN . (6.6) Let us perform basic computations for the general case (6.6).
