Critical 1- and 2-point spin correlations for the O(2) model in 3d bounded domains

We study the critical properties of the 3d O(2) universality class in bounded domains through Monte Carlo simulations of the clock model. We use an improved version of the latter, chosen to minimize finite-size corrections at criticality, with 8 orientations of the spins and in the presence of vacancies. The domain chosen for the simulations is the slab configuration with fixed spins at the boundaries. We obtain the universal critical magnetization profile and two-point correlations, which favorably compare with the predictions of the critical geometry approach based on the Yamabe equation. The main result is that the correlations, once the dimensionful contributions are factored out with the critical magnetization profile, are shown to depend only on the distance between the points computed using a metric found solving the corresponding fractional Yamabe equation. The quantitative comparison with the corresponding results for the Ising model at criticality is shown and discussed. Moreover, from the magnetization profiles the critical exponent η is extracted and found to be in reasonable agreement with up-to-date results.

Local pointwise second derivative estimates for strong solutions to the $$\sigma _k$$-Yamabe equation on Euclidean domains

We prove local pointwise second derivative estimates for positive $$W^{2,p}$$ W 2 , p solutions to the $$\sigma _k$$ σ k -Yamabe equation on Euclidean domains, addressing both the positive and negative cases. Generalisations for augmented Hessian equations are also considered.

Expansions for distributional solutions of the elliptic equation in two dimensions

Assume [Formula: see text] is a planar domain, and [Formula: see text] is a locally bounded distributional solution to the elliptic equation [Formula: see text] where [Formula: see text] is a constant, [Formula: see text] and [Formula: see text] are real analytic functions defined on [Formula: see text] and the real line [Formula: see text], respectively. We establish asymptotic expansions of [Formula: see text] to arbitrary orders near [Formula: see text], which complements the recent results of Han–Li–Li on the Yamabe equation, Guo–Li–Wanon the weighted Yamabe equation, and partly extends that of Guo–Wan–Yang on the Liouville equation in a punctured disc. Our method is a combination of a priori estimate and mathematical induction.

Yamabe systems and optimal partitions on manifolds with symmetries

<p style='text-indent:20px;'>We prove the existence of regular optimal <inline-formula><tex-math id="M1">\begin{document}$ G $\end{document}</tex-math></inline-formula>-invariant partitions, with an arbitrary number <inline-formula><tex-math id="M2">\begin{document}$ \ell\geq 2 $\end{document}</tex-math></inline-formula> of components, for the Yamabe equation on a closed Riemannian manifold <inline-formula><tex-math id="M3">\begin{document}$ (M,g) $\end{document}</tex-math></inline-formula> when <inline-formula><tex-math id="M4">\begin{document}$ G $\end{document}</tex-math></inline-formula> is a compact group of isometries of <inline-formula><tex-math id="M5">\begin{document}$ M $\end{document}</tex-math></inline-formula> with infinite orbits. To this aim, we study a weakly coupled competitive elliptic system of <inline-formula><tex-math id="M6">\begin{document}$ \ell $\end{document}</tex-math></inline-formula> equations, related to the Yamabe equation. We show that this system has a least energy <inline-formula><tex-math id="M7">\begin{document}$ G $\end{document}</tex-math></inline-formula>-invariant solution with nontrivial components and we show that the limit profiles of its components separate spatially as the competition parameter goes to <inline-formula><tex-math id="M8">\begin{document}$ -\infty $\end{document}</tex-math></inline-formula>, giving rise to an optimal partition. For <inline-formula><tex-math id="M9">\begin{document}$ \ell = 2 $\end{document}</tex-math></inline-formula> the optimal partition obtained yields a least energy sign-changing <inline-formula><tex-math id="M10">\begin{document}$ G $\end{document}</tex-math></inline-formula>-invariant solution to the Yamabe equation with precisely two nodal domains.</p>

A Class of Fully Nonlinear Equations Arising in Conformal Geometry

In this paper, we consider the equations of Krylov type in conformal geometry on closed smooth Riemannian manifolds, which can be viewed as an extension of $\sigma _k$-Yamabe equation. Moreover, we prove local gradient and 2nd-derivative estimates for solutions to these equations and establish an existence result.