**
T. Chantladze, N. Kandelaki, A. Lomtatidze, D. Ugulava**

##
On a Generalization of the Dirichlet Integral

**abstract:**

Using the theory of spline functions, we investigate the problem of minimization
of a generalized Dirichlet integral of the form

F_{l}(u) = Integral over
W of (l^{2} + Sum from i=1 to n of u_{xi}^{2}
) ^{p/2}, 1 < p < infinity

where W is a bounded domain of an n-dimensional
Euclidean space R_{n}, l >= 0 is a fixed
number, and u_{xi} is a generalized derivative of the
function u with respect to x_{i} according to Sobolev defined on
W. Minimization is realized with respect to the
functions u whose boundary values on G form a
preassigned function, and for them F_{l}(u)
is finite.