> [!definition] > > Let $U \subset \real^d$ be bounded and [[Open Set|open]] with $\partial U \in C^1$, $L$ be a [[Second Order Linear Partial Differential Operator|second order linear partial differential operator]] with $L^\infty(U)$ coefficients in divergence form, and $f \in L^2(U)$. If $u \in H^2(U)$ is a solution to the boundary-value problem $Lu = f$ and $Tu = 0$, then $u \in H_0^2(U)$ by the [[Zero Trace Theorem|zero trace theorem]]. > > Therefore for any $u$ in the [[Sobolev Space|Sobolev space]] $H_0^1(U)$, $u$ is a **weak solution** to the boundary value problem $Lu = f$ and $Tu = 0$ if $\angles{u, \phi}_L = \angles{v, \phi}_{L^2}$ for all $\phi \in H_0^1(U)$.