论文标题
内部背部问题：Hölder空间中的本地分辨率
The interior Backus problem: local resolution in Hölder spaces
论文作者
论文摘要
我们证明了欧几里得球中靠背内部问题的存在。该问题包括从边界上其梯度模量的知识确定球中的谐波函数。这个问题严重非线性。从物理的角度来看，问题可以解释为从速度场在边界上的模量的测量值确定球内不可压缩和无旋流的速度潜力。线性化问题是一个不规则的倾斜衍生物问题，为此发生了衍生物丧失的现象。结果，通过线性化问题的解决方案变得有问题。在这里，我们将垂直高度溶液周围的问题线性化，并表明（垂直）在垂直方向上（垂直）对称或奇怪的对称的解决方案不会发生衍生物的丢失。然后，基于Hölder空间中的临时加权估计值，因此可行的标准固定点参数是可行的。
We prove an existence result for the Backus interior problem in the Euclidean ball. The problem consists in determining a harmonic function in the ball from the knowledge of the modulus of its gradient on the boundary. The problem is severely nonlinear. From a physical point of view, the problem can be interpreted as the determination of the velocity potential of an incompressible and irrotational fluid inside the ball from measurements of the velocity field's modulus on the boundary. The linearized problem is an irregular oblique derivative problem, for which a phenomenon of loss of derivatives occurs. As a consequence, a solution by linearization of the Backus problem becomes problematic. Here, we linearize the problem around the vertical height solution and show that the loss of derivatives does not occur for solutions which are either (vertically) axially symmetric or oddly symmetric in the vertical direction. A standard fixed point argument is then feasible, based on ad hoc weighted estimates in Hölder spaces.
