多重劣調和函数
L ( φ ) = ( ∂ 2 φ ∂ x 1 2 + ∂ 2 φ ∂ x 2 2 ) [ ( ∂ φ ∂ y 1 ) 2 + ( ∂ φ ∂ y 2 ) 2 ] + ( ∂ 2 φ ∂ y 1 2 + ∂ 2 φ ∂ y 2 2 ) [ ( ∂ φ ∂ x 1 ) 2 + ( ∂ φ ∂ x 2 ) 2 ] − 2 ( ∂ 2 φ ∂ x 1 ∂ y 1 + ∂ 2 φ ∂ x 2 ∂ y 2 ) ( ∂ φ ∂ x 1 ∂ φ ∂ y 1 + ∂ φ ∂ x 2 ∂ φ ∂ y 2 ) − 2 ( ∂ 2 φ ∂ x 1 ∂ y 2 − ∂ 2 φ ∂ x 2 ∂ y 1 ) ( ∂ φ ∂ x 1 ∂ φ ∂ y 2 − ∂ φ ∂ x 2 ∂ φ ∂ y 1 ) {\displaystyle \scriptstyle {\begin{aligned}L(\varphi )&=\left({\frac {\partial ^{2}\varphi }{\partial x_{1}^{2}}}+{\frac {\partial ^{2}\varphi }{\partial x_{2}^{2}}}\right)\left[\left({\frac {\partial \varphi }{\partial y_{1}}}\right)^{2}+\left({\frac {\partial \varphi }{\partial y_{2}}}\right)^{2}\right]+\left({\frac {\partial ^{2}\varphi }{\partial y_{1}^{2}}}+{\frac {\partial ^{2}\varphi }{\partial y_{2}^{2}}}\right)\left[\left({\frac {\partial \varphi }{\partial x_{1}}}\right)^{2}+\left({\frac {\partial \varphi }{\partial x_{2}}}\right)^{2}\right]\\\\&-2\left({\frac {\partial ^{2}\varphi }{\partial x_{1}\partial y_{1}}}+{\frac {\partial ^{2}\varphi }{\partial x_{2}\partial y_{2}}}\right)\left({\frac {\partial \varphi }{\partial x_{1}}}{\frac {\partial \varphi }{\partial y_{1}}}+{\frac {\partial \varphi }{\partial x_{2}}}{\frac {\partial \varphi }{\partial y_{2}}}\right)-2\left({\frac {\partial ^{2}\varphi }{\partial x_{1}\partial y_{2}}}-{\frac {\partial ^{2}\varphi }{\partial x_{2}\partial y_{1}}}\right)\left({\frac {\partial \varphi }{\partial x_{1}}}{\frac {\partial \varphi }{\partial y_{2}}}-{\frac {\partial \varphi }{\partial x_{2}}}{\frac {\partial \varphi }{\partial y_{1}}}\right)\end{aligned}}}