> [!definition] > > Let $L$ be a second order [[Linear Partial Differential Operator|linear partial differential operator]], then $L$ is in **divergence form** when written as > $ > Lf = \sum_{j, k = 1}^d D_k(a^{jk}D_jf) + \sum_{j = 1}^d b^jD_jf + cf > $ > and **non-divergence form** when written as > $ > Lf = \sum_{j, k = 1}^da^{jk}D_jD_kf + \sum_{j = 1}^d b^jD_jf + cf > $ > > The divergence form expression can be converged into a non-divergence form expression without changing the coefficients $\bracsn{a^{jk}}$, and the matrix $(a^{jk})$ can be assumed to be symmetric. > [!definition] > > Let $L$ be a second order partial differential operator with $L^\infty$ coefficients in divergence form, then $L: W^{1, 1}_{\text{loc}} \to \cd'$ is well-defined on all [[Distributional Derivative|weakly differentiable functions]] via > $ > \angles{Lf, \phi}_\cd = \sum_{j, k = 1}^d \angles{a^{jk}D_jf, D_k\phi}_\cd + \sum_{j = 1}^d \angles{b^jD_jf, \phi}_\cd + \angles{cf, \phi}_\cd > $ > Moreover, if $f \in H^1$, then $Lf$ is a [[Bounded Linear Map|bounded linear map]] on the [[Space of Test Functions|test functions]] $H^1 \cap \cd$ with respect to the [[Sobolev Space|Sobolev]] norm, and hence admits a [[Linear Extension Theorem|unique extension]] to $H_0^1$. Since $L: H^1 \to (H_0^1)^*$ is bounded, $L$ induces a bounded sesquilinear map > $ > \angles{Lf, \phi}_L := \sum_{j, k = 1}^d \angles{a^{jk}D_jf, D_k\phi}_{L^2} + \sum_{j = 1}^d \angles{b^jD_jf, \phi}_{L^2} + \angles{cf, \phi}_{L^2} > $ > known as the **sesquilinear form associated with** $L$. > > *Proof*. With simply $L^\infty$ coefficients, $a^{jk}D_jf$ may not even be weakly differentiable. Hence the differentiation $D_j$ takes the distributional derivative, and $L$ is a mapping from $W^{1, 1}$ to $\cd'$. Under this interpretation, > $ > \begin{align*} > \angles{Lf, \phi}_{\cd} &= -\sum_{j, k = 1}^d \angles{D_k(a_{jk}D_jf), \phi}_{\cd} + \sum_{j = 1}^d \angles{b^jD_jf, \phi}_\cd + \angles{cf, \phi}_\cd \\ > &= \sum_{j, k = 1}^d \angles{a_{jk}D_jf, D_k\phi}_{\cd} + \sum_{j = 1}^d \angles{b^jD_jf, \phi}_\cd + \angles{cf, \phi}_\cd > \end{align*} > $ > Since $D^\alpha \overline{\phi} = \overline{D^\alpha \phi}$, the $L^2$ equality follows directly from above, and $\inp_L$ is bounded.