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)