A priori estimates for boundary value elliptic problems via first order systems
A priori estimates for boundary value elliptic problems via first order systems
复制标题
通过一阶系统对边值椭圆问题的先验估计
DOI：
10.1016/j.cyto.2013.06.086
复制
发表时间：
2014
期刊：
影响因子：
--
通讯作者：
Sebastian Stahlhut
中科院分区：
文献类型：
--
作者：
P. Auscher;Sebastian Stahlhut
证明了上半空间中具有垂直独立系数的椭圆型方程或方程组的弱解的一些先验估计.这些估计适用于各种拓扑结构中的Dirichlet和Neumann型边值问题。我们研究的是包括能量解决方案在内的各类解决方案。对于这些解，我们使用由其余法梯度满足的一阶系统和与之相关的Hardy空间理论来描述，但该方法也允许我们设计不一定是能量解的解。我们得到了平方函数、非切极大函数和边界迹范数之间的精确比较。主要研究结果的指数范围与Hardy空间(可以是抽象空间)何时被识别为具有调和分布的具体空间有关。我们考虑了一些适应的非切尖函数，并证明了它与平方函数的比较。我们得到了层势的有界性结果，边界行为，特别是强极限，这是新的，以及跳跃关系。一个应用是可解性的外推“a la{v{S}}ne{u\i}berg”.另一个应用是在不作进一步假设的情况下摄动$L中的系数的可解性的稳定性.我们强调我们的结果不需要De Giorgi-Nash假设，当我们这样做时，我们改进了已有的结果.
We prove a number of \textit{a priori} estimates for weak solutions of elliptic equations or systems with vertically independent coefficients in the upper-half space. These estimates are designed towards applications to boundary value problems of Dirichlet and Neumann type in various topologies. We work in classes of solutions which include the energy solutions. For those solutions, we use a description using the first order systems satisfied by their conormal gradients and the theory of Hardy spaces associated with such systems but the method also allows us to design solutions which are not necessarily energy solutions. We obtain precise comparisons between square functions, non-tangential maximal functions and norms of boundary trace. The main thesis is that the range of exponents for such results is related to when those Hardy spaces (which could be abstract spaces) are identified to concrete spaces of tempered distributions. We consider some adapted non-tangential sharp functions and prove comparisons with square functions. We obtain boundedness results for layer potentials, boundary behavior, in particular strong limits, which is new, and jump relations. One application is an extrapolation for solvability ''\'a la {\v{S}}ne{\u\i}berg". Another one is stability of solvability in perturbing the coefficients in $L^\infty$ without further assumptions. We stress that our results do not require De Giorgi-Nash assumptions, and we improve the available ones when we do so.