Elliptic partial differential equation

at a given point $ x $

A partial differential equation of order $ m $,

$$ \sum a _ {i _ {1} \dots i _ {n} } ( x) \frac{\partial ^ {m} u }{\partial x _ {1} ^ {i _ {1} } \dots \partial x _ {r} ^ {i _ {n} } } + L _ {1} u = f ,\ \ \sum_{j=1}^ { n } i _ {j} = m , $$

such that $ L _ {1} $ is a differential operator of order less than $ m $, whose characteristic equation at $ x $,

$$ K ( \lambda _ {1} \dots \lambda _ {n} ) = \sum a _ {i _ {1} \dots i _ {n} } ( x) \lambda _ {1} ^ {i _ {1} } \dots \lambda _ {n} ^ {i _ {n} } = 0 ,\ \sum _ { j=1}^ { n } i _ {j} = m , $$

has no real roots except $ \lambda _ {1} = 0 \dots \lambda _ {n} = 0 $.

For second-order equations the characteristic form is quadratic,

$$ Q ( \lambda _ {1} \dots \lambda _ {n} ) = \sum _ {i , j = 1 } ^ { n } A _ {ij} ( x) \lambda _ {i} \lambda _ {j} , $$

and can be brought to the form

$$ Q = \sum_{i=1}^ { n } \alpha _ {i} \xi _ {i} ^ {2} $$

by a non-singular affine transformation of the variables $ \lambda _ {i} = \lambda _ {i} ( \xi _ {1} \dots \xi _ {n} ) $, $ i = 1 \dots n $.

When all $ \alpha _ {i} = 1 $ or all $ \alpha _ {i} = - 1 $, the equation is said to be of elliptic type.

A partial differential equation is said to be of elliptic type in its domain of definition if it is elliptic at every point of this domain.

An elliptic partial differential is called uniformly elliptic if there are positive numbers $ k _ {0} $ and $ k _ {1} $ such that

$$ k _ {0} \sum_{i=1}^ { n } \lambda _ {i} ^ {2} \leq Q ( \lambda _ {1} \dots \lambda _ {n} ) \leq k _ {1} \sum_{i=1}^ { n } \lambda _ {i} ^ {2} . $$

For references see Differential equation, partial.

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