Multi-dimensional variational problem

variational problem involving partial derivatives

A problem in the calculus of variations (cf. Variational calculus) in which it is required to determine an extremum of a functional depending on a function of several independent variables. Ordinary variational problems, in which functionals of functions of one independent variable are considered, may be called one-dimensional variational problems, in this sense.

An example of a two-dimensional variational problem is the problem of determining a function of two independent variables, $ u ( x , y ) $, which, together with its first-order partial derivatives, is continuous and yields an extremum of the functional

$$ \tag{1 } I ( u) = {\int\limits \int\limits } _ { D } F ( x , y , u , u _ {x} , u _ {y} ) d x d y $$

under the boundary condition

$$ \tag{2 } u ( x , y ) \mid _ {l} = \ u _ {0} ( x , y ) , $$

where $ l $ is a closed contour bounding a domain $ D $, $ u _ {0} ( x , y ) $ is a given function and $ F ( x , y , u , u _ {x} , u _ {y} ) $ is a twice continuously-differentiable function jointly in its arguments. Let $ u ( x , y ) $ be a solution of the problem (1), (2). Substitution of a comparison function $ u ( x , y ) + \alpha \eta ( x , y ) $, where $ \eta ( x , y ) \mid _ {l} = 0 $ and $ \alpha $ is a numerical parameter, into (1), differentiation with respect to $ \alpha $ and equating $ \alpha = 0 $, gives the following expression for the first variation of the functional:

$$ \tag{3 } \delta I = {\int\limits \int\limits } _ { D } ( F _ {u} \eta + F _ {u _ {x} } \eta _ {x} + F _ {u _ {y} } \eta _ {y} ) d x d y . $$

If $ u ( x , y ) $ has continuous second-order derivatives, then it is easy to show that a necessary condition for $ \delta I $ to vanish is:

$$ \tag{4 } F _ {u} - \frac \partial {\partial x } F _ {u _ {x} } - \frac \partial {\partial y } F _ {u _ {y} } = 0 . $$

Equation (4) is called the Euler–Ostrogradski equation (sometimes the Ostrogradski equation). This equation must be satisfied by a function $ u ( x , y ) $ which gives an extremum of (1) under the boundary conditions (2). The Euler–Ostrogradski equation is analogous to the Euler equation for one-dimensional problems. In expanded form, (4) is a second-order partial differential equation.

In the case of a triple integral and a function $ u ( x , y , z ) $ depending on three independent variables, the Euler–Ostrogradski equation takes the form:

$$ F _ {u} - \frac \partial {\partial x } F _ {u _ {x} } - \frac \partial {\partial y } F _ {u _ {y} } - \frac \partial {\partial z } F _ {u _ {z} } = 0 . $$

The following condition is an analogue of the Legendre condition. In order that $ u ( x , y ) $ gives at least a weak extremum of (1) it is necessary that at each interior point of $ D $,

$$ F _ {u _ {x} u _ {x} } F _ {u _ {y} u _ {y} } - F _ {u _ {x} u _ {y} } ^ { 2 } \geq 0 . $$

For a minimum necessarily $ F _ {u _ {x} u _ {x} } \geq 0 $, and for a maximum $ F _ {u _ {x} u _ {x} } \leq 0 $.

Discontinuous multi-dimensional variational problems have also been considered (see [4]).

References

[1]	R. Courant, D. Hilbert, "Methods of mathematical physics. Partial differential equations" , 2 , Interscience (1965) (Translated from German)
[2]	V.I. Smirnov, "A course of higher mathematics" , 4 , Addison-Wesley (1964) (Translated from Russian)
[3]	N.I. Akhiezer, "The calculus of variations" , Blaisdell (1962) (Translated from Russian)
[4]	M.K. Kerimov, "On two-dimensional continuous problems of variational calculus" Trudy Tbilis. Mat. Inst. Akad. Nauk GruzSSR , 18 (1951) pp. 209–219 (In Russian)