Let (Σ, g) be a compact Riemannian surface without boundary and W 1,2 (Σ) be the usual Sobolev space. For any real number p > 1 and α ∈ R , we define a functional Jα,8π(u) = 1 2 (∫ Σ |∇gu| 2 dvg − α(∫ Σ |u| p dvg) 2/p) − 8π log ∫ Σ he u dvg on a function space H = { u ∈ W 1,2 (Σ) : ∫ Σ udvg = 0 } , where h is a positive smooth function on Σ. Denote λ1,p(Σ) = inf u∈H, ∫ Σ |u| p dvg =1 ∫ Σ |∇gu| 2 dvg. If α < λ1,p(Σ) and Jα,8π has no minimizer on H , then we obtain the exact value of infH Jα,8π by using a method of blow-up analysis. Hence, if infH Jα,8π is not equal to that value, then Jα,8π|H has a critical point that satisfies a Kazdan-Warner equation. This recovers a recent result of Yang and Zhu

The mechanical behaviors of TB991 weld sealant under cyclic loading conditions were experimentally investigated. The evolution of relaxation stress, cyclic softening, and dissipated energy was evaluated with the effect of strain amplitude and mean strain. The experimental results showed that the stress–strain response curves of the first loading‐unloading and cyclic loading‐unloading were significantly different. The phenomenon of stress relaxation and cyclic softening occurred under cyclic strain loading conditions. Furthermore, the relaxation stress and dissipated energy decreased rapidly during the initial cyclic loading and then steadily decreased with the increase of cycle number, while the cyclic softening increased rapidly at first and then steadily. Besides, a viscoelastic constitutive model was proposed which can describe the different shapes of stress–strain curve between the initial loading‐unloading and the cyclic loading‐unloading and also considers the cyclic stress relaxation and cyclic softening of the materials under cyclic loading condition. Comparisons between the numerical results and the experimental data demonstrated that the proposed model can better describe the mechanical behavior of TB991 weld sealant under cyclic loading conditions.

The adsorption of cadmium from simulated mining wastewater by coal waste (CW) and calcinationmodified coal waste (MCW) was investigated. Effects of pH, initial concentration, particle size of adsorbent, adsorbent dosage and temperature were studied in batch experiments. The adsorption efficiency for cadmium increased with increasing pH, and the optimum pH for cadmium adsorption onto MCW and CW was 6.0 and 6.5, respectively. Kinetic experiments showed that the adsorption equilibrium was reached within 120 min and followed pseudo-second-order model well. The adsorption isotherm data fit Langmuir and Freundlich models, and the adsorption capacity of cadmium on the two adsorbents increased with increasing temperature from 298 to 318 K. MCW had a higher adsorption capacity of cadmium than CW, because calcination treatment can make CW to have more loose structure and higher specific surface area. Thermodynamic parameters, the Gibbs free energy change (DG 0 ), enthalpy change (DH 0 ) and entropy change (DS 0 ), were calculated and the results showed that the adsorption of cadmium on CW and MCW was spontaneous and endothermic. Fourier transform infrared studies indicated silanol and aluminol groups were responsible for cadmium binding. The desorption results indicated that the two adsorbents could be used repeatedly at least three times without significant decrease in the adsorption capacity for cadmium. The results suggested that modified CW could have high potential as low-cost adsorbent for cadmium removal.

In this paper, we establish several trace Trudinger-Moser inequalities and obtain the corresponding extremals on a compact Riemann surface (Σ, g) with smooth boundary ∂Σ. To be exact, let λ 1 (∂Σ) denotes the first eigenvalue of the Laplace-Beltrami operator ∆g on ∂Σ. Moreover, for any 0 ≤ α < λ 1 (∂Σ), we set H = {u ∈ W 1,2 (Σ, g) : Σ |∇gu| 2 dvg − α ∂Σ u 2 dsg 1/2 ≤ 1 and ∂Σ u dsg = 0}, where W 1,2 (Σ, g) is the usual Sobolev space. By the method of blow-up analysis, we first prove the supremum sup u∈H ∂Σ e πu 2 dsg is attained by some function uα ∈ H ∩ C ∞ Σ. Further, we extend the result to the case of higher order eigenvalues. The results generalize those of Li-Liu [9] and Yang [19, 20].

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