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Zbl 0729.35011
Mantlik, Frank
Partial differential operators depending analytically on a parameter. (English)
[J] Ann. Inst. Fourier 41, No. 3, 577-599 (1991). ISSN 0373-0956; ISSN 1777-5310/e

Let $P(\lambda,D)=\sum\sb{\vert \alpha \vert \le m}a\sb{\alpha}(\lambda)D\sp{\alpha}$ be a differential operator with constant coefficients $a\sb{\alpha}$ depending analytically on a parameter $\lambda$. Assume that the family $\{$ P($\lambda$,D)$\}$ is of constant strength. We investigate the equation $P(\lambda,D)f\sb{\lambda}\equiv g\sb{\lambda}$ where $g\sb{\lambda}$ is a given analytic function of $\lambda$ with values in some space of distributions and the solution $f\sb{\lambda}$ 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.
[F.Mantlik (Dortmund)]

MSC 2000:: *35B30 Dependence of solutions of PDE on initial and boundary data
35E05 Fundamental solutions (PDE with constant coefficients)

Keywords: linear differential operator; analytic dependence; elementary solutions; constant coefficients; constant strength; regular fundamental solution

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