PDEs Second-order elliptic equations generalize the Laplace and Poisson equations: where typically models the density of some quantity of interest in some open and bounded domain with . is an Elliptic Differential Operator Misplaced &-\sum_{i=1}^n \sum_{j=1}^n \partial_{x_j} \left( A^{ij}(x) \partial_{x_i} u \right) + \sum_{i=1}^n b^i(x) \partial_{x_i} u + c(x) u & \text{divergence form}\\ \\ -\sum_{i=1}^n \sum_{j=1}^n A^{ij}(x) \partial_{x_i x_j} u + \sum_{i=1}^n b^i(x) \partial_{x_i} u + c(x) u & \text{non-divergence form}\\ \end{cases}$$ if the matrix $A^{ij}(x)$ is uniformly **elliptic** a.e: $$\left\langle y, A(x)y \right\rangle = \sum_{i=1}^n \sum_{j=1}^n A^{ij}(x) y_i y_j \geq \theta ||y||^2$$ for some $\theta > 0$ and $\forall \space y \in \mathbb{R}^n$. - $A^{ij}$ represents the (an)isoptropic diffusion of $u$ throughout $U$ - $b^i$ represents the transport of $u$ throughout $U$ - $c$ represents local increase or decrease of $u$ in $U$ - A general elliptic differential operator may fail to be invertible, so that regularization provides the correct [[Variational Principle for Elliptic Equations]] for the divergence form - Existence and uniqueness is guaranteed in certain cases by the fact that elliptic differential operators can be extended to [[Fredholm Operator]]s - [[Regularity of Elliptic Equations]] also follows from being Fredholm - The non-divergence form prevents use of integration by parts, so that existence of solutions follows from [[Maximum Principles for Elliptic Equations]]