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Positivity for equations involving polyharmonic operators with Dirichlet boundary conditions. (English) Zbl 0892.35031
Based on Boggio’s formula the authors derive estimates for Green’s functions of polyharmonic operators in balls $$B$$. These estimates are delicate and require a subtle analysis. Different types of maximum principles are then derived. At first, boundary value problems of the type $((-\triangle)^m+A)u=f \text{ in }B, \;D_mu=0 \text{ on } \partial B,$ where $$A$$ is a perturbation of the polyharmonic operator, are considered. Conditions are given which insure that a positive $$f$$ implies that the solution is also positive. The same question is discussed for systems of equations. The use of the Neumann series plays a crucial role.

##### MSC:
 35B50 Maximum principles in context of PDEs 35J40 Boundary value problems for higher-order elliptic equations 35C15 Integral representations of solutions to PDEs
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