The N U D problem

TL;DR

This paper investigates the hybrid spectral problem with mixed Dirichlet and Neumann boundary conditions, proposing a conjecture for a specific coefficient and deriving the conformal determinant for a 2-disc with semi-circular boundary regions.

Contribution

It introduces a conjecture for the C_1 coefficient in the hybrid spectral problem and derives the conformal determinant for a 2-disc with mixed boundary conditions.

Findings

Proposed a conjecture for the C_1 coefficient.

Derived the conformal determinant on a 2-disc with semi-circular boundary regions.

Discussed implications for higher coefficients.

Abstract

The hybrid spectral problem where the field satisfies Dirichlet conditions (D) on part of the boundary of the relevant domain and Neumann (N) on the remainder is discussed in simple terms. A conjecture for the C_1 coefficient is presented and the conformal determinant on a 2-disc, where the D and N regions are semi-circles, is derived. Comments on higher coefficients are made.