A positivity preserving property result for the biharmonic operator under partially hinged boundary conditions
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A positivity preserving property result for the biharmonic operator under partially hinged boundary conditions Elvise Berchio1 · Alessio Falocchi1 Received: 19 August 2020 / Accepted: 4 November 2020 © The Author(s) 2020
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. Keywords Biharmonic · Positivity preserving · Partially hinged plate · Green function Mathematics Subject Classification 35G15 · 35J08 · 35B09
1 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]. Nevertheless, 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). As a matter of example, let us consider the clamped plate problem: { 2 Δ u=f in Ω (1) u = |∇u| = 0 on 𝜕Ω where Ω ⊂ ℝn is a bounded domain and f ∈ L2 (Ω) ; we say that the above problem satisfies the PPP if the following implication holds
f ⩾ 0 in Ω
⇒
u ⩾ 0 in Ω ,
* Alessio Falocchi [email protected] 1
Dipartimento di Scienze Matematiche, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Turin, Italy
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where u is a (weak) solution to (1). The validity of the PPP generally depends either on the choice of the boundary conditions or on the geometry of the domain. For instance, 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. Things become somehow simpler if in (1), instead of the Dirichlet boundary conditions, we take the Navier boundary conditions, i.e., we consider the hinged plate problem: { 2 Δ u=f in Ω u = Δu = 0 on 𝜕Ω. Here, the PPP follows by applying twice the comparison principle for the Laplacian under Dirichlet boundary conditions. It is worth noticing that smoothness of the domain cannot be overlooked since it has been shown by Nazarov and Sweers [15] that, also in this case, the PPP may fail for planar domains with an interior corner. We refer to the book [9] for more details and PPP results under different kinds of boundary conditions, e.g., Steklov boundary conditions, and to [10–12, 16–19, 21] for up-to-date results on the topic. In the present paper, we focus on the less studie
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