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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 where Ω is a bounded domain of an 𝑛-dimensional Euclidean space 𝑅𝑛, λ ≥ 0 is a fixed number, and 𝑢𝑥𝒊 is a generalized according to Sobolev with respect to 𝑥𝒊 derivative of the function 𝑢 defined on Ω. Minimization is realized with respect to the functions 𝑢 whose boundary values on Γ form a preassigned function, and for them 𝐹λ(𝑢) is finite.