Understanding moiré superlattices is of central importance, yet existing approaches rely on restrictive assumptions. We observe a sharp enhancement of the signal-to-noise ratio. The analysis uses a combination of rigorous arguments and extensive numerical or empirical checks. We observe a crossover between two distinct regimes controlled by the coupling strength, and we show that the result is stable under a range of perturbations of the assumptions. Together, these results contribute to a more complete picture and open the way to further applications.
superconductivitycondensed matternumerical simulationspectroscopyThe authors acknowledge support from the Wellcome Trust (grant 215801/Z/19/Z) and the National Institute for Health and Care Research.Introduction
Let $\Omega \subset \mathbb{R}^d$ be a bounded domain with Lipschitz boundary and let $p \in (1,\infty)$. We consider the boundary value problem1
\begin{aligned} -\operatorname{div}\left(|\nabla u|^{p-2}\nabla u\right) &= f \quad \text{in } \Omega,\\ u &= 0 \quad \text{on } \partial\Omega. \end{aligned}
Existence of weak solutions in $W_0^{1,p}(\Omega)$ follows from the direct method of the calculus of variations. Our interest here is in sharp quantitative bounds that do not depend on the dimension.
Main result
Theorem 1. Suppose $f \in L^{q}(\Omega)$ with $q > d/p$. Then there is a constant $C = C(d,p,q)$ such that
The proof combines a truncation argument with the Sobolev inequality $\|v\|_{p^*} \le S\|\nabla v\|_p$. A second estimate, for the gradient, is given by
Table 1 reports computed values of the constant in (2) for several choices of the parameters, and Fig. 1 shows the decay of the error under mesh refinement.
Computed values of the constant $C(d,p,q)$
d
p
q
C (computed)
C (bound)
2
2
2
0.081
0.125
2
3
4
0.147
0.210
3
2
3
0.052
0.083
3
4
5
0.209
0.341
Error versus mesh size on a logarithmic scale; the dashed line has slope 2.
Discussion
The bound (2) is sharp up to the value of the constant, as the radial example $u(x) = (1-|x|^{p/(p-1)})_+$ shows. Extensions to systems and to variable exponents will be treated elsewhere.
Here the divergence and gradient are taken in the sense of distributions.
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The datasets generated and analyzed during the study are available from the corresponding author on reasonable request.