Extension and lacunas of solutions of linear partial differential equations. (English) Zbl 0853.35022
Summary: Let \(K\subset Q\) be compact, convex sets in \(\mathbb{R}^n\) with \({\overset\circ K}\not=\emptyset\) and let \(P(D)\) be a linear, constant coefficient PDO. It is characterized in various ways when each zero solution of \(P(D)\) in the space \({\mathcal E}(K)\) of all \(C^\infty\)-functions on \(K\) extends to a zero solution in \({\mathcal E}(Q)\) resp. in \({\mathcal E}(\mathbb{R}^n)\). The most relevant characterizations are in terms of Phragmén-Lindelöf conditions on the zero variety of \(P\) in \(\mathbb{C}^n\) and in terms of fundamental solutions for \(P(D)\) with lacunas.
MSC:
| 35E05 | Fundamental solutions to PDEs and systems of PDEs with constant coefficients |
| 35B60 | Continuation and prolongation of solutions to PDEs |
| 32U05 | Plurisubharmonic functions and generalizations |
| 46F05 | Topological linear spaces of test functions, distributions and ultradistributions |
Keywords:
Whitney extension of zero-solutions; Phragmén-Lindelöf conditions for algebraic varieties; fundamental solutions with lacunas; continuous linear right inverses for constants coefficient partial differential operatorsReferences:
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