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On positivity for the biharmonic operator under Steklov boundary conditions. (English) Zbl 1155.35019
Let us consider \(\Omega\) being a bounded smooth domain in \(\mathbb R^n,\) \(n \geq 2, \) and the following elliptic problem, called Steklov problem, having parameter in the boundary condition:
\[ \Delta^2 u= f\quad\text{in }\Omega, \qquad u=f\text{ and }\Delta u= \alpha u_\nu\quad \text{on }\partial \Omega \]
where \(\alpha \in C(\partial \Omega),\) \(f \in L^2(\Omega)\) and \(\nu\) is the outside normal.
The authors are interested in conditions on \(\alpha \) which guarantee that the above Steklov problem is positivity preserving, meaning \(f \geq 0 \) implies that \(u \geq 0.\) It is proved that this property is associated to the parameter considered in the boundary condition. Gazzola and Sweers point out that for Navier and Dirichlet boundary conditions it is known that all \(\alpha \) for which the positivity preserving property represent an interval. This result is similar to that is true obtained for the Steklov problem.

MSC:
35J40 Boundary value problems for higher-order elliptic equations
31B30 Biharmonic and polyharmonic equations and functions in higher dimensions
31B20 Boundary value and inverse problems for harmonic functions in higher dimensions
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
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