# zbMATH — the first resource for mathematics

Partial differential operators depending analytically on a parameter. (English) Zbl 0729.35011
Let $$P(\lambda,D)=\sum_{| \alpha | \leq m}a_{\alpha}(\lambda)D^{\alpha}$$ be a differential operator with constant coefficients $$a_{\alpha}$$ depending analytically on a parameter $$\lambda$$. Assume that the family $$\{$$ P($$\lambda$$,D)$$\}$$ is of constant strength. We investigate the equation $$P(\lambda,D)f_{\lambda}\equiv g_{\lambda}$$ where $$g_{\lambda}$$ is a given analytic function of $$\lambda$$ with values in some space of distributions and the solution $$f_{\lambda}$$ is required to depend analytically on $$\lambda$$, too. As a special case we obtain a regular fundamental solution of P($$\lambda$$,D) which depends analytically on $$\lambda$$. This result answers a question of L. Hörmander.

##### MSC:
 35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs 35E05 Fundamental solutions to PDEs and systems of PDEs with constant coefficients
Full Text: