Strong unique continuation and global regularity estimates for nanoplates

TL;DR

This paper investigates a sixth order elliptic operator in nanoplates, establishing well-posedness, regularity estimates, and strong unique continuation properties using Carleman estimates, advancing mathematical understanding in strain gradient elasticity.

Contribution

It introduces new regularity and unique continuation results for a sixth order elliptic operator in nanoplates, with rigorous proofs and applications to isotropic materials.

Findings

Established well-posedness of the Neumann problem.

Derived interior and boundary regularity estimates.

Proved strong unique continuation properties using Carleman estimates.

Abstract

In this paper we analyze some properties of a sixth order elliptic operator arising in the framework of the strain gradient linear elasticity theory for nanoplates in flexural deformation. We first rigorously deduce the weak formulation of the underlying Neumann problem as well as its well posedness. Under some suitable smoothness assumptions on the coefficients and on the geometry we derive interior and boundary regularity estimates for the solution of the Neumann problem. Finally, for the case of isotropic materials, we obtain new Strong Unique Continuation results in the interior, in the form of doubling inequality and three spheres inequality, by a Carlemann estimates approach.