The objective of this work is an existence proof for variational solutions u to parabolic minimizing problems. Here, the functions being considered are defined on a metric measure space \(({\mathcal {X}}, d, \mu )\). For such parabolic minimizers that coincide with Cauchy-Dirichlet data \(\eta \) on the parabolic boundary of a space-time-cylinder \(\varOmega \times (0, T)\) with an open subset \(\varOmega \subset {\mathcal {X}}\) and \(T > 0\), we prove existence in the parabolic Newtonian space \(L^p(0, T; {\mathcal {N}}^{1,p}(\varOmega ))\). In this paper we generalize results from Collins and Herán (Nonlinear Anal 176:56–83, 2018) where only time-independent Cauchy–Dirichlet data have been considered. We argue completely on a variational level.

这项工作的目的是变解的存在性证明ü抛物线最小化的问题。在此，要考虑的功能是在度量度量空间\（（{{mathcal {X}}，d，\ mu）\）上定义的。对于这样的抛物线形极小子，它与带有开放子集\（\ varOmega \的时空圆柱\（\ varOmega \ times（0，T）\）的抛物线边界上的Cauchy-Dirichlet数据\（\ eta \）子集{\ mathcal {X}} \）和\（T> 0 \），我们证明存在于抛物牛顿空间\（L ^ p（0，T; {\ mathcal {N}} ^ {1，p} （\ varOmega））\）。在本文中，我们概括了Collins和Herán（非线性分析176：56-83，2018年）的结果，其中仅考虑了时间无关的柯西-狄利克雷数据。我们完全在变体层面上争论。