Quasi-periodic Poincaré–Bertrand formula

Let $\gamma_0$ denote the portion of $\gamma$ lying in the strip $[-d/2,d/2) \times \mathbb{R}$ and define

$$\mathcal{H}_\xi[\sigma](\boldsymbol{x}) = \int_{\gamma_0} \partial_{\boldsymbol \tau(\boldsymbol{y})} G_{\xi,L}(\boldsymbol{x}-\boldsymbol{y})\sigma(\boldsymbol{y})\,\mathrm{d}\boldsymbol{y}, \qquad \mathcal{D}_\xi[\sigma](\boldsymbol{x}) = \int_{\gamma_0} \partial_{\boldsymbol n(\boldsymbol{y})} G_{\xi,L}(\boldsymbol{x}-\boldsymbol{y})\sigma(\boldsymbol{y})\,\mathrm{d}\boldsymbol{y}.$$

These are the quasi-periodic Hilbert transform and Laplace double-layer potential, respectively. If $\sigma$ is Hölder continuous and quasi-periodic, then Quasi-periodic Poincaré–Bertrand formula.

$$\frac14\mathcal{H}_\xi^2[\sigma] = -\frac{\sigma}{4} + \mathcal{D}_\xi^2[\sigma].$$

This conjecture proposes the quasi-periodic analogue of the standard Poincaré–Bertrand identity, which is used to simplify boundary integral formulations for free-space flexural scattering problems. Its validity is needed for the subsequent derivation of the quasi-periodic integral equations and is not established in the source.