紧黎曼流形上的椭圆边界值问题

讨论一类非齐次非线性椭圆边界值问题.利用极大值原理证明了该问题解的梯度估计.作为它的应用得到了解的效率比估计.     

Big,Strong, Neutral,Twisted, and Chiral π Acids

General synthetic access to expanded π‐acidic surfaces of variable size, topology, chirality, and π acidity is reported. The availability of π surfaces with these characteristics is essential to develop the functional relevance of anion–π interactions with regard to molecular recognition, translocation, and transformation. The problem is that, with expanded π surfaces, the impact of electron‐withdrawing substituents decreases and the high π acidity needed for strong anion–π interactions can be more difficult to obtain. To overcome this problem, it is herein proposed to build large surfaces from smaller fragments and connect these fragments with bridges that are composed only of single atoms. Two central surfaces for powerful anion–π interactions, namely, perfluoroarenes and naphthalenediimides (NDIs), were selected as fragments and coupled with through sulfide bridges. Their oxidation to sulfoxides and sulfones, as well as fluorine substitution in the peripheral rings, provides access to the full chemical space of relevant π acidities. According to cyclic voltammetry, LUMO levels range from ?3.96 to ?4.72 eV. With sulfoxide bridges, stereogenic centers are introduced to further enrich the intrinsic planar chirality of the expanded surfaces. The stereoisomers were separated by chiral HPLC and characterized by X‐ray crystallography. Their topologies range from chairs to π boats, and the latter are reminiscent of the cation–π boxes in operational neuronal receptors. With pentafluorophenyl acceptors, the π acidity of NDIs with two sulfoxide groups in the core reaches ?4.45 eV, whereas two sulfone moieties give a value of ?4.72 eV, which is as low as with four ethyl sulfone groups, that is, a π superacid near the limit of existence. Beyond anion–π interactions, these conceptually innovative π‐acidic surfaces are also of interest as electron transporters in conductive materials.