= Uniformly elliptic operator
= Uniformly elliptic
{synonym}
= Uniform ellipticity
{synonym}
For a real second-order principal coefficient matrix $A(x)=(a^{ij}(x))$, <uniform ellipticity> means that one constant $\lambda>0$, independent of $x$, satisfies
$$
\xi^TA(x)\xi\ge\lambda|\xi|^2\qquad\hbox{for every }\xi\in\mathbb R^n.
$$
For measurable coefficients the condition is imposed <almost everywhere>; for a classical pointwise operator it holds at every relevant point. The <quadratic form> depends only on the <symmetric part> of $A$. Many estimates also assume a finite uniform upper bound or bounded full coefficients; the lower bound alone does not supply them.
For example, $A=\operatorname{diag}(1,4)$ has lower bound one. On $0<x_1<1$, the matrix $A(x)=\operatorname{diag}(x_1^2,1)$ is positive definite at each point but has no positive uniform lower bound: test $\xi=(1,0)$ as $x_1\downarrow0$. On the restricted strip $x_1>\delta>0$, a bound $\min(\delta^2,1)$ works. This distinguishes a <strictly elliptic operator> from a <uniformly elliptic operator> on the whole domain.
In a <divergence-form elliptic operator>, the same inequality controls the principal <gradient> energy. Together with bounded coefficients and the <Poincaré inequality>, it underlies <coercivity> and weak existence. A nonsymmetric coefficient matrix needs separate boundedness control, as shown by <quadratic ellipticity does not bound a nonsymmetric coefficient matrix>.<Partial Differential Equations by John K. Hunter>[Chapter 4, Definition 4.16 and Examples 4.17–4.18]{ref}
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