论文标题
生物分子静电中的非线性泊松托玻璃器方程的弱配方
Weak formulations of the nonlinear Poisson-Boltzmann equation in biomolecular electrostatics
论文作者
论文摘要
我们考虑在生物大分子的静电模型的背景下，我们考虑了非线性泊松玻璃到人方程，该方程嵌入了一个有界域中，该结构域中包含一个任意数量的离子物种的溶液，不一定是电荷中性的。所得的半线性椭圆方程结合了几个困难：非线性占离子迁移率的指数生长和缺乏符号保存，测量分子内部电荷产生的数据，以及整个分子边界的不连续拼写。利用建模假设，即点源和非线性在域的不相交部分上有效，可以将电势的线性分解为常规和单数组件。可以用于常规部分，但无限的非线性使相应的功能在sobolev空间中不可差。通过证明最小化器的界限，这些与常规组件的标准$ h^1 $弱配方有关，并且在Boccardo和Gallouët的框架中具有全部潜力。最后，这类型的弱解决方案的独特性对于更通用的半线性问题，测量数据验证了策略，因为所考虑的不同分解和测试空间必须导致相同的解决方案。
We consider the nonlinear Poisson-Boltzmann equation in the context of electrostatic models for a biological macromolecule, embedded in a bounded domain containing a solution of an arbitrary number of ionic species which is not necessarily charge neutral. The resulting semilinear elliptic equation combines several difficulties: exponential growth and lack of sign preservation in the nonlinearity accounting for ion mobility, measure data arising from point charges inside the molecule, and discontinuous permittivities across the molecule boundary. Exploiting the modelling assumption that the point sources and the nonlinearity are active on disjoint parts of the domain, one can use a linear decomposition of the potential into regular and singular components. A variational argument can be used for the regular part, but the unbounded nonlinearity makes the corresponding functional not differentiable in Sobolev spaces. By proving boundedness of minimizers, these are related to standard $H^1$ weak formulations for the regular component and in the framework of Boccardo and Gallouët for the full potential. Finally, a result of uniqueness of this type of weak solutions for more general semilinear problems with measure data validates the strategy, since the different decompositions and test spaces considered must then lead to the same solution.
