Higher order boundary value problems model the equilibrium states of rods and plates, amongst other things. For this reason, they received a lot of attention in the twentieth century, but there are still a lot of “big problems” to solve. In particular, the problem as to whether or not a given Green’s operator is positivity preserving has proven to be a very difficult problem to investigate. Indeed, the current situation is pretty bad. The only known examples of positivity preserving Green’s operators for this simple boundary value problem are discs, small perturbations of discs, and a family of limacons. This is striking for the aforementioned boundary value problem is one of the simplest high-order boundary value problems involving an elliptic differential operator that one can consider. In short, the current state of affairs seems to be (i) some Green’s operators are positivity preserving; (ii) some are not positivity preserving; and (iii) only God knows why one is of one type instead of the other.

If you are interested in the theory of positivity preservation and higher order boundary value problems that involve elliptic differential operators and homogeneous Dirichlet boundary conditions, I suggest starting with the survey article [Sweers 2012]. If you want a more advanced treatment of the subject I suggest reading the monograph [Gazzola, Grunau, and Sweers 2010]. Much of the recent history of the subject has to do with a program of bounding the negative part of a given Green’s functions from below and comparing it to the positive part. This is what is done in the ground-breaking paper [Grunau and Robert 2010]. At this point —the year 2024 — this research program has become highly developed, and it suggests that even with all the negative results that have characterized research about positivity preservation and higher order boundary value problems, one has it that for many higher order boundary value problems the Green’s function is “almost positive”. That is, the negative part of the Green’s function is small compared to the positive part.

Even with the above results in place, the theory of positivity preservation and higher order boundary value problems involving elliptic differential operators could still use some more attention. In [Grunau and Sweers 2014], it is shown that the solution of the higher order boundary value problem exhibited here can be sign changing. I was shocked when I first encountered this result, because I immediately realized its implications. I.e., for some values of x in U, the integral of the problem’s Green’s function G(x,y) over U will be negative, and there are positive Green’s operators that can take a positive function to one that has a negative mean value, I turned these realizations into a paper, and I made it available here. (We will cite this paper as [Raske 2024]). It suggests that it would be useful to have a classification system for Green’s operators based on how far from being positivity preserving they are. Thus, like the aforementioned papers, it employs the notion of something being “almost positive”, but the arguments used and the results obtained are completely different.

Bibliography:

[Gazzola, Grunau, and Sweers 2010]: F. Gazzola, H.C. Grunau, and G. Sweers. Polyharmonic boundary value problems, Positivity preserving and nonlinear higher order elliptic equations in bounded domains.
Springer Lecture Notes in Mathematics 1991, Springer-Verlag: Heidelberg etc., 2010.

[Grunau and Robert 2010], H.-Ch. Grunau and F. Robert. Positivity and almost positivity of biharmonic Green's functions under Dirichlet boundary conditions. Arch. Rational Mech. Anal., 195, 865-898 (2010).

[Grunau and Sweers 2014] H.C. Grunau and G. Sweers, G. A clamped plate with a uniform weight may change sign.
Discrete Cont. Dynam. Systems - S (Proceedings etc.) 7, 761 - 766 (2014).

[Raske 2024] D. Raske. Positivity and Green’s operators.arXiv:2408.1746v4 [math.AP].

[Sweers 2016] G. Sweers, On sign preservation for clotheslines, curtain rods, elastic membranes and thin plates., Jahresber. Dtsch. Math.-Ver. 118 (2016), 275–320.

(Note: All of these papers (except the one by David Raske) can be found, free, on the authors’ academic websites.)