Un'osservazione sulla risolubilità formale delle equazioni alle derivate parziali lineari a coefficienti costanti

Bratti, Giuliano

Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 5 (1994) no. 1, pp. 23-25

Résumé

Let \( s \) be a formal series and \( p = p(D) = \sum_{\alpha} a_{\alpha}D^{\alpha} \) a partial differential operator, with constant coefficients. We say that \( s \) is a formal solution of \( ps = 0 \) if there exists \( f \in C^{\infty} (\mathbb{R}^{n}) \) such that \( \sum_{q} \left[ f^{(q)} (0) / q! \right] z^{q} = s \) and \( (p(D) f)^{(q)}(0) = 0\), \( \forall q \in N^{n} (z = (z_{1}, \ldots , z_{n}) \). We give a characterization of formal solvability of \( p \) in terms of extension, like distributions, over all \( \mathbb{R}^{n} \) of the elements of the space \( \{ u \in D' (\mathbb{R}^{n} —\{0\} ): p(-D)u = 0 \} \).