In this paper, we consider a variational formulation for the Dirichlet problem of the wave equation with zero boundary and initial conditions, where we use integration by parts in space and time. To prove unique solvability in a subspace of $H^1(Q$) with $Q$ being the space-time domain, the classical assumption is to consider the right-hand side $f$ in $L^2(Q)$. Here, we analyze a generalized setting of this variational formulation, which allows us to prove unique solvability also for $f$ being in the dual space of the test space, i.e., the solution operator is an isomorphism between the ansatz space and the dual of the test space. This new approach is based on a suitable extension of the ansatz space to include the information of the differential operator of the wave equation at the initial time $t=0$. These results are of utmost importance for the formulation and numerical analysis of unconditionally stable space-time finite element methods, and for the numerical analysis of boundary element methods to overcome the well-known norm gap in the analysis of boundary integral operators.

翻译：在本文中,我们考虑对零边界和初始条件的波形Drichlet问题的变式配方。 要证明在1美元1美元(Q$)的子空间和1美元(Q$)的时空域中独有的溶解性, 典型的假设是考虑右侧的美元(f) 美元(L2美元)(Q) 。 在这里, 我们分析这种变式配方的概括性设置, 使我们能够证明在试验空间的双重空间里也有独特的溶解性, 也就是说, 溶解操作器是ansatz空间和试验空间的双层之间的一种不形态。 这一新办法的基础是适当扩展asatz空间, 以包括最初时间波形方形不同操作器操作器的信息 $t=0美元。 这些结果对于无条件稳定的空间定时要素方法的拟订和数字分析, 以及对边界要素方法的数字分析, 以克服边界整体操作器分析中众所周知的规范差距, 至关重要。