Regularity results of the thin obstacle problem for the p(x)-Laplacian

We study thin obstacle problems involving the energy functional with $p(x)$-growth. We prove higher integrability and Hölder regularity for the gradient of minimizers of the thin obstacle problems under the assumption that the variable exponent $p(x)$ is Hölder continuous.     

Nonlinear obstacle problems with double phase in the borderline case

In this paper we study a double phase problem with an irregular obstacle. The energy functional under consideration is characterized by the fact that both ellipticity and growth switch between a type of polynomial and a type of logarithm, which can be regarded as a borderline case of the double phase functional with $(p,q)$-growth. We obtain an optimal global Calderón–Zygmund type estimate for the obstacle problem with double phase in the borderline case.     

Global gradient estimates for non-uniformly elliptic equations

We consider a nonlinear and non-uniformly elliptic problem in divergence form on a bounded domain. The problem under consideration is characterized by the fact that its ellipticity rate and growth radically change with the position, which provides a model for describing a feature of strongly anisotropic materials. We establish the global Calderón–Zygmund type estimates for the distributional solution in the case that the boundary of the domain is of class \(C^{1,\beta }\) for some \(\beta >0\).     

Chrysin Derivative CM1 and Exhibited Anti-Inflammatory Action by Upregulating Toll-Interacting Protein Expression in Lipopolysaccharide-Stimulated RAW264.7 Macrophage Cells

Although our previous study revealed that gamma-irradiated chrysin enhanced anti-inflammatory activity compared to intact chrysin, it remains unclear whether the chrysin derivative, CM1, produced by gamma irradiation, negatively regulates toll-like receptor (TLR) signaling. In this study, we investigated the molecular basis for the downregulation of TLR4 signal transduction by CM1 in macrophages. We initially determined the appropriate concentration of CM1 and found no cellular toxicity below 2 μg/mL. Upon stimulation with lipopolysaccharide (LPS), CM1 modulated LPS-stimulated inflammatory action by suppressing the release of proinflammatory mediators (cytokines TNF-α and IL-6) and nitric oxide (NO) and downregulated the mitogen-activated protein kinase (MAPK) and nuclear factor-κB (NF-κB) signaling pathways. Furthermore, CM1 markedly elevated the expression of the TLR negative regulator toll-interacting protein (Tollip) in dose- and time-dependent manners. LPS-induced expression of cell surface molecules (CD80, CD86, and MHC class I/II), proinflammatory cytokines (TNF-α and IL-6), COX-2, and iNOS-mediated NO were inhibited by CM1; these effects were prevented by the knockdown of Tollip expression. Additionally, CM1 did not affect the downregulation of LPS-induced expression of MAPKs and NF-κB signaling in Tollip-downregulated cells. These findings provide insight into effective therapeutic intervention of inflammatory disease by increasing the understanding of the negative regulation of TLR signaling induced by CM1.     

A Family of Molecular Sieves Containing Framework‐Bound Organic Structure‐Directing Agents

Organic structure‐directing agents (OSDAs), such as quaternary ammonium cations and amines, used in the synthesis of zeolites and related crystalline microporous oxides usually end up entrapped inside the void spaces of the crystallized inorganic host lattice. But none of them is known to form direct chemical bonds to the framework of these industrially important catalysts and adsorbents. We demonstrate that ECR‐40, currently regarded as a typical silicoaluminophosphate molecular sieve, constitutes instead a new family of inorganic‐organic hybrid networks in which the OSDAs are covalently bonded to the inorganic framework. ECR‐40 crystallization begins with the formation of an Al–OSDA complex in the liquid phase in which the Al is octahedrally coordinated. This unit is incorporated in the crystallizing ECR‐40. Subsequent removal of framework‐bound OSDAs generates Al‐O‐Al linkages in a fully tetrahedrally coordinated framework.     

Gradient estimates for elliptic systems with measurable coefficients in nonsmooth domains

We consider an elliptic system in divergence form with measurable coefficients in a nonsmooth bounded domain to find a minimal regularity requirement on the coefficients and a lower level of geometric assumption on the boundary of the domain for a global W ^1,p , 1 < p < ∞, regularity. It is proved that such a W ^1,p regularity is still available under the assumption that the coefficients are merely measurable in one variable and have small BMO semi-norms in the other variables while the domain can be locally approximated by a hyperplane, a so called δ-Reifenberg domain, which is beyond the Lipschitz category. This regularity easily extends to a certain Orlicz-Sobolev space.