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On a generalization of the Dirichlet integral. (English) Zbl 1128.49300
Using the theory of spline functions the authors investigate the problem of minimization of a generalized Dirichlet integral $F_{\lambda}(u)=\int_{\Omega}\left(\lambda^2+\sum_{i=1}^n u_{x_i}^2 \right)^{p/2}d\sigma,\qquad1<p<\infty,$ where $$\Omega$$ is a bounded domain of an $$n$$-dimensional Euclidean space $$\mathbb R^n$$, $$\lambda\geq0$$ is a fixed number, and $$u_{x_i}$$ is a generalized according to Sobolev with respect to $$x_i$$ derivative of the function $$u$$ defined on $$\Omega$$. Minimization is realized with respect to the functions $$u$$ whose boundary values on $$\Gamma$$ form a preassigned function, and for them $$F_{\lambda}(u)$$ is finite.
##### MSC:
 49J10 Existence theories for free problems in two or more independent variables 35J20 Variational methods for second-order elliptic equations 41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
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