楕円型偏微分方程式
L u = ∑ i = 1 n ∑ j = 1 n a i , j ∂ 2 u ∂ x i ∂ x j + (lower-order terms) = 0 {\displaystyle Lu=\sum _{i=1}^{n}\sum _{j=1}^{n}a_{i,j}{\frac {\partial ^{2}u}{\partial x_{i}\partial x_{j}}}\quad {\text{ + (lower-order terms)}}=0\,} ,
ここで、L は楕円型作用素である。
例えば、三次元 (x,y,z) においては
a ∂ 2 u ∂ x 2 + b ∂ 2 u ∂ x ∂ y + c ∂ 2 u ∂ y 2 + d ∂ 2 u ∂ y ∂ z + e ∂ 2 u ∂ z 2 + (lower-order terms) = 0 , {\displaystyle a{\frac {\partial ^{2}u}{\partial x^{2}}}+b{\frac {\partial ^{2}u}{\partial x\partial y}}+c{\frac {\partial ^{2}u}{\partial y^{2}}}+d{\frac {\partial ^{2}u}{\partial y\partial z}}+e{\frac {\partial ^{2}u}{\partial z^{2}}}{\text{ + (lower-order terms)}}=0,}