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

TL;DR

This paper establishes a boundary doubling inequality for solutions to the Kirchhoff-Love plate equation with Dirichlet conditions, which implies strong unique continuation at the boundary and relaxes conditions for Cauchy problem uniqueness.

Contribution

It introduces a new boundary doubling inequality for the Kirchhoff-Love equation using Carleman estimates and solution reflection, extending previous boundary unique continuation results.

Findings

Proves a boundary doubling inequality for Kirchhoff-Love solutions

Establishes strong unique continuation property at the boundary

Weakens conditions for Cauchy problem uniqueness

Abstract

The main result of this paper is a doubling inequality at the boundary for solutions to the Kirchhoff-Love isotropic plate's equation satisfying homogeneous Dirichlet conditions. This result, like the three sphere inequality with optimal exponent at the boundary proved in Alessandrini, Rosset, Vessella, Arch. Ration. Mech. Anal. (2019), implies the Strong Unique Continuation Property at the Boundary (SUCPB). Our approach is based on a suitable Carleman estimate, and involves an ad hoc reflection of the solution. We also give a simple application of our main result, by weakening the standard hypotheses ensuring uniqueness for the Cauchy problem for the plate equation.