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Positivity properties of elliptic boundary value problems of higher order. (English) Zbl 0894.35016
From the authors’ introduction: “As the simple polyharmonic function \(x\to-| x|^2\) demonstrates, strong maximum principles are obviously false in higher-order elliptic equations. But it is reasonable to ask whether in the Dirichlet problem \[ Lu= f\quad\text{in }\Omega,\quad (-\partial/\partial\nu)^ju|_{\partial\Omega}= \phi|_{\partial\Omega}\quad\text{for }j= 0,1,\dots,m- 1, \] positive data yield positive solutions. Numerous counterexamples show that there is in general no affirmative answer to our positivity question. So the appropriate question is as follows: are there suitable conditions on the operator \(L\), the domain \(\Omega\) and the choices of boundary data to be prescribed homogeneously, such that positive data yield positive solutions? In the present note, we focus on the role of the boundary conditions”.
Reviewer: M.Chicco (Genova)

MSC:
35B50 Maximum principles in context of PDEs
35J40 Boundary value problems for higher-order elliptic equations
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