Optimal three spheres inequality at the boundary for the Kirchhoff-Love plate's equation with Dirichlet conditions

TL;DR

This paper establishes an optimal boundary three sphere inequality for Kirchhoff-Love plate equations, leading to a strong unique continuation property at the boundary, using Carleman estimates and conformal mappings.

Contribution

It provides the first nontrivial strong unique continuation property at the boundary for fourth-order equations with bi-Laplacian principal part.

Findings

Proved a boundary three sphere inequality with optimal exponent.

Established the Strong Unique Continuation Property at the Boundary (SUCPB).

Developed a novel approach using conformal mappings and reflection techniques.

Abstract

We prove a three sphere inequality with optimal exponent at the boundary for solutions to the Kirchhoff-Love plate's equation satisfying homogeneous Dirichlet conditions. This result implies the Strong Unique Continuation Property at the Boundary (SUCPB). Our approach is based on the method of Carleman estimates, and involves the construction of an ad hoc conformal mapping preserving the structure of the operator and the employment of a suitable reflection of the solution with respect to the flatten boundary which ensures the needed regularity of the extended solution. To the authors' knowledge, this is the first (nontrivial) SUCPB result for fourth-order equations with bi-Laplacian principal part.