A positivity preserving property result for the biharmonic operator under partially hinged boundary conditions

Elvise Berchio,Alessio Falocchi

doi:10.1007/s10231-020-01054-6

Elvise Berchio, Alessio Falocchi

Open Access

https://doi.org/10.1007/s10231-020-01054-6

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Abstract

It is well known that for higher order elliptic equations, the positivity preserving property (PPP) may fail. In striking contrast to what happens under Dirichlet boundary conditions, we prove that the PPP holds for the biharmonic operator on rectangular domains under partially hinged boundary conditions, i.e., nonnegative loads yield positive solutions. The result follows by fine estimates of the Fourier expansion of the corresponding Green function.

Highlights

One of the main obstructions in the development of the theory of higher order elliptic equations is represented by the loss of general maximum principles, see, e.g., [9, Chapter 1]
Due to the central role that these technical tolls play in the general theory of second-order elliptic equations, in the last century a large part of literature has focused in studying whether the related boundary-value problems possibly enjoy the so-called positivity preserving property (PPP in the following)
From the seminal works by Boggio [5, 6], it is known that problem (1) satisfies the PPP when Ω is a ball in Rn, while, in [7], Coffman and Duffin proved that the PPP does not hold when Ω is a two-dimensional domain containing a right angle, such as a square or a rectangle

Summary

Introduction

One of the main obstructions in the development of the theory of higher order elliptic equations is represented by the loss of general maximum principles, see, e.g., [9, Chapter 1]. In the article we develop an accurate analysis of the qualitative properties of the m and we show, in particular, that they are strictly decreasing with respect to m ∈ N+ This monotonicity issue is achieved by studying the sign of the derivatives of the m ; since they have highly involved analytic expressions, in order to detect their sign, we set up a clever scheme where, step by step, we cancel out the dependence of some variables through optimization arguments, see Remark 4.1 of Sect.

Notations and main results

Green function computation