Surjektivität und Parameterabhängigkeit bei partiellen Differentialoperatoren. (Surjectivity and parameter dependence in partial differential operators) (Q580695)

Partial differential equations \[ (1)\quad P(x,D_ x)\quad u(x,t)=f(x,t) \] are investigated, where the operator \(P(x,D_ x)\) is elliptic of degree \(m\geq 1\) with analytic coefficients in the variable x and independent of the variable t. To study the solvability of (1), t is considered a parameter; the methods used include existence and regularity theorems for elliptic equations, Sobolev embedding theorems and results on parameter dependent equations [cf. the author, J. Reine Angew. Math. 309, 55-85 (1979; Zbl 0407.46060) and the author and \textit{D. Vogt}, Manuscr. Math. 32, 1-27 (1980; Zbl 0456.46058)]. The main results of the article are the following: Let \(\Omega_ 1\subseteq {\mathbb{R}}^ N\) and \(\Omega_ 2\subseteq {\mathbb{R}}^ M\) be open; then for any continuous \(f\in C(\Omega_ 1\times \Omega_ 2)\), (1) has a solution \(u\in C(\Omega_ 1\times \Omega_ 2)\). If \(P(D_ x)\) has constant coefficients, this even holds over \(\tilde P(D)\)-convex open sets \(\Omega \subseteq {\mathbb{R}}^{N+M}\); here \(\tilde P\) is the polynomial \(\tilde P(\xi,\tau) = P(\xi)\) in \((N+M)\) variables. Similar assertions hold for \(C^ r\)-functions; for \(1\leq r<m\) (and for \(r=\infty)\), \(C^ r\)-solutions are obtained, while for \(m\leq r<\infty\) some differentiability is lost. It was shown by \textit{L. C. Piccinini} [Boll. Unione Mat. Ital., IV. Ser. 7, 12-28 (1973; Zbl 0264.35003)], that for real-analytic functions \(f\in {\mathcal A}(\Omega_ 1\times \Omega_ 2)\), (1) need not have solutions in \({\mathcal A}(\Omega_ 1\times \Omega_ 2)\). However, it is shown here that for \(f\in {\mathcal A}(\Omega_ 2,{\mathcal A}(\Omega_ 1))\) (\(\subsetneqq {\mathcal A}(\Omega_ 1\times \Omega_ 2)!)\), a solution \(u\in {\mathcal A}(\Omega_ 2,{\mathcal A}(\Omega_ 1))\) of (1) exists.