Doubling Inequality and Nodal Sets for Solutions of Bi-Laplace Equations

Abstract

We investigate the doubling inequality and nodal sets for the solutions of bi-Laplace equations. A polynomial upper bound for the nodal sets of solutions and their gradient is obtained based on the recent development of nodal sets for Laplace eigenfunctions by Logunov. In addition, we derive an implicit upper bound for the nodal sets of solutions. We show two types of doubling inequalities for the solutions of bi-Laplace equations. As a consequence, the rate of vanishing is given for the solutions.2010 Mathematics Subject Classification. 35J05, 35J30, 58J50, 35J05.

Upper bounds of nodal sets for eigenfunctions of eigenvalue problems

“…Proof. The proof of the Corollary follows from the arguments in Corollary 1 in [Zh4]. For the completeness of the presentation, we present the proof.…”

“…Such weight function φ(x) was introduced by Hörmander in [H1]. The following quantitative Carleman estimates were established in [Zh4] for bi-Laplace operators.…”

“…(2.27) Similar arguments have also been carried out in Lemma 4. Interested readers may refer to Lemma 4 in the paper or [Zh4] for details of the argument.…”

Doubling Inequality at the Boundary for the Kirchhoff -- Love Plate's Equation with Dirichlet Conditions

“…In this Appendix we prove Carleman estimate (3.28). We proceed, similarly to [16], [31], [40], in a standard way by iterating a suitable Carleman estimate for the Laplace operator.…”

“…We emphasize that, with respect to the Carleman estimate employed in [8], the presence of the first term in the left hand side of (1.4) is the key ingredient in order to prove our doubling inequality at the boundary. At the best of our knowledge, Bakri is the first author who derived a doubling inequality in the interior starting from a Carleman estimate of the kind (1.4) [10], see also [11] and [40].…”