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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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