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We establish some sharp affine weighted L 2 Sobolev inequalities on the upper half space, which involves a divergent operator with degeneracy on the boundary. Moreover, for some certain exponents cases, we also characterize the extremal functions and best constants. Our approach only relies on the L 2 structure of gradient norm, affine invariance and a class of weighted L 2 Sobolev inequality on the upper half space. This is a simple approach which does not depend on the geometric structure of Euclidean space such as Brunn–Minkowski theory on convex geometry.

In this article, we establish some reverse weighted Hardy-Littlewood-Sobolev inequalities on the Heisenberg group. We then show the existence of extremal functions for the above inequalities by combining the subcritical approach and the renormalization method.

We establish some sharp affine weighted L2 Sobolev inequalities on the upper half space, which involves a divergent operator with degeneracy on the boundary. Moreover, for some certain exponents cases, we also characterize the extremal functions and best constants. Our approach only relies on the L2 structure of gradient norm, affine invariance and a class of weighted L2 Sobolev inequality on the upper half space. This is a simple approach which does not depend on the geometric structure of Euclidean space such as Brunn–Minkowski theory on convex geometry. [ABSTRACT FROM AUTHOR]

BACKGROUND: Active and smart food packaging based on natural polymers and pH‐sensitive dyes as indicators has attracted widespread attention. In the present study, an antioxidant and amine‐response color indicator film was developed by incorporating bilirubin (BIL) into the κ‐carrageenan (Carr) matrix. RESULTS: It was found that the introduction of BIL had no effect on the crystal/chemical structure, water sensitivity and mechanical performance of the Carr‐based films. However, the barrier properties to light and the thermal stability were significantly improved after the addition BIL. The Carr/BIL composite films exhibited excellent 1,1‐diphenyl‐2‐picryl‐hydrazyl (i.e. DPPH)/2,2′‐azino‐bis‐(3‐ethylbenzothiazoline‐6‐sulfonic acid) (i.e. ABTS) free radical scavenging abilities and color responsiveness to different concentrations of ammonia. The application assay reflected that the Carr/BIL0.0075 film was effective in delaying the oxidative deterioration of shrimp during storage and realizing the color response of its freshness through the change of b* value. CONCLUSION: Active and smart packaging films were successfully prepared by incorporating different contents of BIL into the Carr matrix. The present study helps to further encourage the design and development of a multi‐functional packaging material. © 2023 Society of Chemical Industry. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the asymptotic radial symmetry and monotonicity of positive solutions for weighted fractional parabolic equations in the unit ball. Firstly, we prove the main ingredients of the method-the narrow region principle and asymptotic strong maximum principle for antisymmetric functions in bounded domains. Then, we utilize an asymptotic method of moving planes to obtain asymptotic radial symmetry and monotonicity of positive solutions in the unit ball. [ABSTRACT FROM AUTHOR]

In this paper, we consider the weighted integral system with negative exponents on the upper half space $ \mathbb{R}^{n+1}_+ $ as follows$ \begin{equation*} \begin{cases} u(X) = \displaystyle{\int}_{\mathbb{R}^{n+1}_+}\frac{f(u, v)(Y)}{t^\alpha z^\beta|X-Y|^\lambda}dY, &X\in\mathbb{R}^{n+1}_+, \\ v(X) = \displaystyle{\int}_{\mathbb{R}^{n+1}_+}\frac{g(u, v)(Y)}{ t^\beta z^\alpha|X-Y|^\lambda}dY, &X\in\mathbb{R}^{n+1}_+, \end{cases} \end{equation*} $where $ \alpha, \beta\le0 $, $ \lambda<0 $ and $ X = (x, t), \, Y = (y, z). $ Under the natural conditions on $ f $ and $ g $, we obtain the classification and symmetry of positive solutions by the method of moving spheres in integral forms. Moreover, we generalize our results to integral system on $ \mathbb{R}^{n+m} $. [ABSTRACT FROM AUTHOR]