论文标题
迹线和分歧定理的理论
A theory of traces and the divergence theorem
论文作者
论文摘要
我们在某个功能空间上将其视为线性连续函数的迹线引入了一种通用方法，在该空间中，我们专注于该空间的一些特殊选择。这导致了一个积分的积分，用于计算可集成函数的精确代表和sobolev或BV函数的痕迹。对于具有分布差异的可集成矢量场，我们还可以在任意的鲍尔集集中获得高斯绿色公式。事实证明，通常需要第二个边界积分。积分计算的优点是，不需要在边界上的正常场和痕量函数。高斯绿色公式也可用于Sobolev和BV功能。最后，对于任何开放设置，边界价值问题的弱解决方案的存在作为痕量理论的应用显示。
We introduce a general approach to traces that we consider as linear continuous functionals on some function space where we focus on some special choices for that space. This leads to an integral calculus for the computation of the precise representative of an integrable function and of the trace of a Sobolev or BV function. For integrable vector fields with distributional divergence being a measure, we also obtain Gauss-Green formulas on arbitrary Borel sets. It turns out that a second boundary integral is needed in general. The advantage of the integral calculus is that neither a normal field nor a trace function on the boundary is needed. The Gauss-Green formulas are also available for Sobolev and BV functions. Finally, for any open set the existence of a weak solution of a boundary value problem is shown as application of the trace theory.
