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In this paper, for d ≥ 1 and s ∈ (0 , d 2) , we study the Bianchi–Egnell quotient Q (f) = inf f ∈ H ˙ s (R d) \ B ‖ (- Δ) s / 2 f ‖ L 2 (R d) 2 - S d , s ‖ f ‖ L 2 d d - 2 s (R d) 2 dist H ˙ s (R d) (f , B) 2 , f ∈ H ˙ s (R d) \ B , where S d , s is the best Sobolev constant and B is the manifold of Sobolev optimizers. By a fine asymptotic analysis, we prove that when d = 1 , there is a neighborhood of B on which the quotient Q (f) is larger than the lowest value attainable by sequences converging to B . This behavior is surprising because it is contrary to the situation in dimension d ≥ 2 described recently in König (Bull Lond Math Soc 55(4):2070–2075, 2023). This leads us to conjecture that for d = 1 , Q (f) has no minimizer on H ˙ s (R d) \ B , which again would be contrary to the situation in d ≥ 2 . As a complement of the above, we study a family of test functions which interpolates between one and two Talenti bubbles, for every d ≥ 1 . For d ≥ 2 , this family yields an alternative proof of the main result of König (Bull Lond Math Soc 55(4):2070–2075, 2023). For d = 1 we make some numerical observations which support the conjecture stated above. [ABSTRACT FROM AUTHOR]

For degree ± 1 harmonic maps from R 2 (or S 2 ) to S 2 , Bernand-Mantel et al. (Arch Ration Mech Anal 239(1):219–299, 2021) recently establish a uniformly quantitative stability estimate. Namely, for any map u : R 2 → S 2 with degree ± 1 , the discrepancy of its Dirichlet energy and 4 π can linearly control the H ˙ 1 -difference of u from the set of degree ± 1 harmonic maps. Whether a similar estimate holds for harmonic maps with a higher degree is unknown. In this paper, we prove that a similar quantitative stability result for a higher degree is true only in a local sense. Namely, given a harmonic map, a similar estimate holds if u is already sufficiently near to it (modulo Möbius transforms) and the bound in general depends on the given harmonic map. More importantly, we thoroughly investigate an example of the degree 2 case, which shows that it fails to have a uniformly quantitative estimate like the degree ± 1 case. This phenomenon shows the striking difference between degree ± 1 ones and higher degree ones. Finally, we also conjecture a new uniformly quantitative stability estimate based on our computation. [ABSTRACT FROM AUTHOR]

The curvature estimates for k curvature equations with general right hand sides is a longstanding problem. In this paper, we give a proof of curvature estimates for n < 2 k base on a conjecture of a key inequality. At last, we proved the conjecture is true for k = n - 2 . [ABSTRACT FROM AUTHOR]

In this paper, we investigate the issue of maximizing the total equilibrium population with respect to resources distribution m(x) and diffusion rates d under the prescribed total amount of resources in a logistic model with nonlocal dispersals. Among other things, we show that for d ≥ 1 , there exist C 0 , C 1 > 0 , depending on ‖ m ‖ L 1 only, such that C 0 d ≤ supremum of total population ≤ C 1 d. However, when replaced by random diffusion, a conjecture, proposed by Wei-Ming Ni and justified in Bai et al. (Proc. Am. Math. Soc. 144:2161–2170, 2016), indicates that in the one-dimensional case, supremum of total population = 3 ‖ m ‖ L 1. This reflects serious discrepancies between models with local and nonlocal dispersal strategies. Furthermore, we provide an equivalent characterization about the combination of resource distribution and diffusion rate such that the corresponding total population could reach the optimal order d as d goes to infinity. [ABSTRACT FROM AUTHOR]

Let A and A ∗ be two non-degenerate spherical annuli in R n equipped with the Euclidean metric and the weighted metric | y | 1 - n , respectively. Let F (A , A ∗) denote the class of homeomorphisms h from A onto A ∗ in the Sobolev space W 1 , n - 1 (A , A ∗) . For n = 3 , the second author (Kalaj in Adv Calc Var, 10.1515/acv-2018-0074, 2019) proved that the minimizers of the Dirichlet-type energy E [ h ] = ∫ A ‖ D h (x) ‖ n - 1 | h (x) | n - 1 d x are certain generalized radial diffeomorphisms, where h ∈ F (A , A ∗) . For the case n ≥ 4 , he conjectured that the minimizers are also certain generalized radial diffeomorphisms between A and A ∗ . The main aim of this paper is to consider this conjecture. First, we investigate the minimality of the following combined energy integral: E [ a , b ] [ h ] = ∫ A a 2 ϱ n - 1 (x) ‖ D S (x) ‖ n - 1 + b 2 | ∇ ϱ (x) | n - 1 ϱ n - 1 (x) d x , where h = ϱ S ∈ F (A , A ∗) , ϱ = | h | and a , b > 0 . The obtained result is a generalization of [Kalaj (Adv Calc Var, 10.1515/acv-2018-0074, 2019), Theorem 1.1]. As an application, we show that the above conjecture is almost true for the case n ≥ 4 , i.e., the minimizer of the energy integral E [ h ] does not exist but there exists a minimizing sequence which belongs to the generalized radial mappings. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the structure and stability of the isotropic-nematic interface in 1-D. In the absence of the anisotropic energy, the uniaxial solution is the only global minimizer. In the presence of the anisotropic energy, the uniaxial solution with the homeotropic anchoring is stable for $$L_2<0$$ and unstable for $$L_2>0$$ . We also present many interesting open questions, some of which are related to De Giorgi conjecture. [ABSTRACT FROM AUTHOR]

The goal of this paper is to study the following non-local superlinear elliptic problem (-Δ)su=|u|p-σϕ1inΩ,u=0inRN\Ω,u>0inΩ,where (-Δ)s is the fractional Laplace operator, Ω⊂RN is an open domain with Lipschitz boundary, σ>0, p∈(1,2s∗-1) with 2s∗=2NN-2s and ϕ1 is the first positive eigenfunction of the fractional Laplacian with Dirichlet boundary condition. We prove the non-local version of a conjecture due to Lazer and McKenna (Proc R Soc Edinb Sect A 95:275-283, 1983) by constructing solutions with sharp peaks near local maximum points of ϕ1. [ABSTRACT FROM AUTHOR]