Positivity properties for the clamped plate boundary problem on the ellipse and strip

Abstract

The positivity preserving property for the biharmonic operator with Dirichlet boundary condition is investigated. We discuss here the case where the domain is an ellipse (that may degenerate to a strip) and the data is a polynomial function. We provide various conditions for which the positivity is preserved.

Fractional Laplacians on ellipsoids

“…To argue the optimality and the consistency of our approach, we remark that the root of the mapping a → h(a, 2) can be computed numerically, and it is given by b 0 ≈ 18.94281916344395 (see Figure 3), which is the same threshold found in [28,Theorem 5.2], obtained with different arguments than ours in the study of the bilaplacian in two-dimensional ellipses. Proof of Corollary 1.7.…”

“…iv) A thorough analysis for s = n = 2 is performed in [28], finding a counterexample in terms of a polynomial of degree 6 in an ellipse with axes ratio equal to √ 19 ≈ 4.359. The authors also show that it is not possible to construct a counterexample in an ellipse with polynomials of degree less than 6; moreover, it is also shown that counterexamples with degree 6 polynomials are only possible if the axes ratio is larger than ≈ 4.352 (this threshold also appears in our analysis, see Section 4.1).…”