A second-order differential equation can still fail to control a function. The operator may contain second derivatives, yet only differentiate in some directions. Elliptic operators are the class of second-order operators whose principal part sees every direction, so boundary value problems behave like equilibrium problems rather than transport problems.

The central question is: when does $Lu=f$ determine an unknown from boundary data, support energy estimates, and improve the regularity of solutions? The answer lies in the quadratic form attached to the highest-order coefficients.

[example: A Second Derivative That Misses a Direction] Let $U=(0,1)^2\subsetneq\mathbb R^2$ and define $Lu=-\partial_{x_1}^2u$. For \begin{align*} u(x_1,x_2)&=\sin(2\pi x_2), \end{align*} the function does not depend on $x_1$, so \begin{align*} \partial_{x_1}u(x_1,x_2)&=0,\\ \partial_{x_1}^2u(x_1,x_2)&=0,\\ Lu(x_1,x_2)&=-\partial_{x_1}^2u(x_1,x_2)=0 \end{align*} for every $(x_1,x_2)\in U$. At the same time, \begin{align*} \partial_{x_2}u(x_1,x_2)&=2\pi\cos(2\pi x_2), \end{align*} so the solution can vary in the $x_2$ direction while the operator does not detect that variation. The principal matrix for the second-order part is \begin{align*} A&=\begin{pmatrix}1&0\\0&0\end{pmatrix}. \end{align*} For $\xi=(\xi_1,\xi_2)\in\mathbb R^2$, \begin{align*} A\xi&=\begin{pmatrix}1&0\\0&0\end{pmatrix}\begin{pmatrix}\xi_1\\ \xi_2\end{pmatrix} =\begin{pmatrix}\xi_1\\0\end{pmatrix},\\ \xi^\top A\xi&=\begin{pmatrix}\xi_1&\xi_2\end{pmatrix}\begin{pmatrix}\xi_1\\0\end{pmatrix} =\xi_1^2+\xi_2\cdot 0 =\xi_1^2. \end{align*} If $\xi=(0,\xi_2)$ with $\xi_2\ne0$, then $\xi\ne0$ but \begin{align*} \xi^\top A\xi&=0^2=0. \end{align*} Thus the formula contains a second derivative, but its principal quadratic form loses the whole $x_2$ direction. [/example]

The example shows why ellipticity is a condition on the principal part, not on the visual appearance of the formula. Before a boundary value problem can be estimated, the second-order coefficients must be tested against arbitrary directions. The following definition names the basic pointwise positivity condition.

[definition: Elliptic Matrix Field] Let $U\subsetneq\mathbb R^n$ be open. A measurable matrix field $A:U\to\mathbb R^{n\times n}$ is elliptic if for $\mathcal L^n$-a.e. $x\in U$ and every $\xi\in\mathbb R^n\setminus\{0\}$, \begin{align*} \sum_{i,j=1}^n A_{ij}(x)\xi_i\xi_j&>0. \end{align*} [/definition]

Pointwise positivity prevents a fixed direction from disappearing at almost every fixed point. Estimates over a domain need a quantitative lower bound that does not collapse from point to point, so the primary definition records the uniform version.

## Definition

Pointwise ellipticity still allows the quadratic form to become arbitrarily small as $x$ varies. That is too weak for estimates, because an energy bound must use one constant across the whole region being tested. Uniform ellipticity isolates the quantitative condition that prevents the operator from becoming nearly degenerate on smaller and smaller sets.

[definition: Uniformly Elliptic Matrix Field] Let $U\subsetneq\mathbb R^n$ be open. A measurable matrix field $A:U\to\mathbb R^{n\times n}$ is uniformly elliptic on $U$ if there exists $\theta>0$ such that for $\mathcal L^n$-a.e. $x\in U$ and every $\xi\in\mathbb R^n$, \begin{align*} \sum_{i,j=1}^n A_{ij}(x)\xi_i\xi_j&\ge \theta |\xi|^2. \end{align*} [/definition]

The constant $\theta$ is the ellipticity constant. When $A$ is also bounded, the principal quadratic form is comparable with the Euclidean form $|\xi|^2$, which is why the Laplacian becomes the model operator.

## Operator Forms

### Divergence and Nondivergence Forms

Weak solutions are built from [integration by parts](/theorems/2098), so the coefficients must be arranged in a form compatible with test functions and weak derivatives. This motivates the divergence form operator.

[definition: Divergence Form Elliptic Operator] Let $U\subsetneq\mathbb R^n$ be open. Let $a_{ij},b_i,c\in L^\infty(U)$ for $1\le i,j\le n$, and assume $A=(a_{ij})$ is uniformly elliptic on $U$. The divergence form elliptic operator with these coefficients is the distribution-valued operator \begin{align*} L:H_0^1(U)&\to H^{-1}(U) \end{align*} defined by \begin{align*} Lu&=-\sum_{i,j=1}^n\partial_{x_i}\left(a_{ij}\partial_{x_j}u\right)+\sum_{i=1}^n b_i\partial_{x_i}u+c u \end{align*} in $\mathcal D'(U)$. [/definition]

Divergence form is adapted to [integration by parts](/theorems/210) because derivatives can be transferred onto test functions. In maximum principles, pointwise comparison, and classical regularity estimates, however, the equation is read at a point and the coefficients act directly on the Hessian. That viewpoint cannot be recovered from integration by parts when coefficients are merely rough, so it has to be recorded as a separate form of operator.

[definition: Nondivergence Form Elliptic Operator] Let $U\subsetneq\mathbb R^n$ be open. Let $a_{ij},b_i,c\in L^\infty(U)$ for $1\le i,j\le n$, and assume $A=(a_{ij})$ is uniformly elliptic on $U$. The nondivergence form elliptic operator with these coefficients is the operator \begin{align*} L:C^2(U)&\to L^1_{\mathrm{loc}}(U) \end{align*} defined $\mathcal L^n$-a.e. by \begin{align*} Lu&=-\sum_{i,j=1}^n a_{ij}\partial_{x_i}\partial_{x_j}u+\sum_{i=1}^n b_i\partial_{x_i}u+c u. \end{align*} [/definition]

The two forms agree for constant coefficients. For variable coefficients they support different notions of solution, so a page about elliptic operators must keep both available.

### The Laplacian as Model

[example: The Laplacian] Let $L=-\Delta$ on an [open set](/page/Open%20Set) $U\subsetneq\mathbb R^n$. In nondivergence form this means the second-order coefficient matrix is the identity matrix, so \begin{align*} a_{ij}&=\delta_{ij},\\ Lu&=-\sum_{i,j=1}^n \delta_{ij}\partial_{x_i}\partial_{x_j}u =-\sum_{i=1}^n \partial_{x_i}\partial_{x_i}u =-\sum_{i=1}^n\partial_{x_i}^2u. \end{align*} For every $\xi=(\xi_1,\dots,\xi_n)\in\mathbb R^n$, the principal quadratic form is \begin{align*} \sum_{i,j=1}^n\delta_{ij}\xi_i\xi_j &=\sum_{i=1}^n \delta_{ii}\xi_i\xi_i+\sum_{\substack{1\le i,j\le n\\ i\ne j}}\delta_{ij}\xi_i\xi_j\\ &=\sum_{i=1}^n \xi_i^2+\sum_{\substack{1\le i,j\le n\\ i\ne j}}0\cdot \xi_i\xi_j\\ &=\sum_{i=1}^n \xi_i^2\\ &=|\xi|^2. \end{align*} Thus \begin{align*} \sum_{i,j=1}^n\delta_{ij}\xi_i\xi_j&\ge 1\cdot |\xi|^2 \end{align*} for every $\xi\in\mathbb R^n$, so $-\Delta$ is uniformly elliptic with ellipticity constant $1$. In divergence form the principal part is \begin{align*} -\sum_{i,j=1}^n\partial_{x_i}\left(\delta_{ij}\partial_{x_j}u\right) &=-\sum_{i=1}^n\partial_{x_i}\left(\partial_{x_i}u\right) =-\sum_{i=1}^n\partial_{x_i}^2u. \end{align*} With no first-order or zero-order terms, the associated [bilinear form](/page/Bilinear%20Form) satisfies, for $u\in H_0^1(U)$, \begin{align*} B[u,u] &=\int_U\sum_{i,j=1}^n\delta_{ij}\partial_{x_j}u\,\partial_{x_i}u\,d\mathcal L^n\\ &=\int_U\sum_{i=1}^n \partial_{x_i}u\,\partial_{x_i}u\,d\mathcal L^n\\ &=\int_U\sum_{i=1}^n |\partial_{x_i}u|^2\,d\mathcal L^n\\ &=\int_U|\nabla u|^2\,d\mathcal L^n. \end{align*} Thus the matrix test and the variational test say the same thing: the identity principal matrix charges every coordinate direction, and the energy records exactly the full squared gradient. [/example]

The Laplacian supplies the Euclidean benchmark. General elliptic operators ask which parts of harmonic function theory survive after replacing $|\xi|^2$ by a uniformly comparable variable quadratic form.