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.

在本文中，我们考虑了具有零边界和初始条件的波动方程的 Dirichlet 问题的变分公式，其中我们使用空间和时间的部分积分。证明子空间的唯一可解性$H^1(问$）与Q是所述空间-时间域，经典的假设是考虑右手侧˚F在${升}^2(问)$. 在这里，我们分析了这个变分公式的广义设置，这使我们能够证明f在测试空间的对偶空间中的唯一可解性，即，解算符是 ansatz 空间和测试对偶之间的同构空间。这种新方法基于对 ansatz 空间的适当扩展，以包括初始时间波动方程的微分算子的信息$吨=0$. 这些结果对于无条件稳定时空有限元方法的公式化和数值分析，以及边界元方法的数值分析以克服边界积分算子分析中众所周知的范数差距至关重要。