On the lower bound of the principal eigenvalue of a nonlinear operator

We prove sharp lower bound estimates for the first nonzero eigenvalue of the non-linear elliptic diffusion operator $$L_p$$ L p on a smooth metric measure space, without boundary or with a convex boundary and Neumann boundary condition, satisfying $$BE(\kappa ,N)$$ B E ( κ , N ) for $$\kappa \ne 0$$ κ ≠ 0 . Our results extends the work of Koerber Valtorta (Calc Vari Partial Differ Equ. 57(2), 49 2018) for case $$\kappa =0$$ κ = 0 and Naber–Valtorta (Math Z 277(3–4):867–891, 2014) for the p-Laplacian.

Local Regularity for the Harmonic Map and Yang–Mills Heat Flows

We establish new local regularity results for the harmonic map and Yang–Mills heat flows on Riemannian manifolds of dimension greater than 2 and 4, respectively, obtaining criteria for the smooth local extensibility of these flows. As a corollary, we obtain new characterisations of singularity formation and use this to obtain a local estimate on the Hausdorff measure of the singular sets of these flows at the first singular time. Finally, we show that smooth blow-ups at rapidly forming singularities of these flows are necessarily nontrivial and admit a positive lower bound on their heat ball energies. These results crucially depend on some local monotonicity formulæ for these flows recently established by Ecker (Calc Var Partial Differ Equ 23(1):67–81, 2005) and the Afuni (Calc Var 555(1):1–14, 2016; Adv Calc Var 12(2):135–156, 2019).

Fast and slow decay solutions for supercritical fractional elliptic problems in exterior domains

We consider the fractional elliptic problem: where B1 is the unit ball in ℝ N , N ⩾ 3, s ∈ (0, 1) and p > (N + 2s)/(N − 2s). We prove that this problem has infinitely many solutions with slow decay O(|x|−2s/(p−1)) at infinity. In addition, for each s ∈ (0, 1) there exists P s > (N + 2s)/(N − 2s), for any (N + 2s)/(N − 2s) < p < P s , the above problem has a solution with fast decay O(|x|2s−N). This result is the extension of the work by Dávila, del Pino, Musso and Wei (2008, Calc. Var. Partial Differ. Equ. 32, no. 4, 453–480) to the fractional case.

Complete Willmore Legendrian surfaces in $${\mathbb {S}}^5$$ are minimal Legendrian surfaces

In this paper, we continue to consider Willmore Legendrian surfaces and csL Willmore surfaces in $${\mathbb {S}}^5$$ S 5 , notions introduced by Luo (Calc Var Partial Differ Equ 56, Art. 86, 19, 2017. 10.1007/s00526-017-1183-z). We will prove that every complete Willmore Legendrian surface in $${\mathbb {S}}^5$$ S 5 is minimal and find nontrivial examples of csL Willmore surfaces in $${\mathbb {S}}^5$$ S 5 .