Abstract
The differential properties of the vector extremals of convex functionals of linear growth, depending only on the modulus of the gradient of the desired function, are investigated. It is proved that if the integrand is strictly convex and its derivative is concave for large values of the argument, then, under some additional conditions, the generalized solution is regular in an open set of full measure. Another result consists in the fact that if the integrand is linear for all large values of the argument, then, under some additional conditions, there exists an open set in which the gradient of the solution is continuous and its modulus is strictly smaller than the value of the parameter, starting from which the integrand becomes a linear function.
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Translated from Problemy Matematicheskogo Analiza, No. 11, pp. 51–79, 1990.
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Seregin, G.A. Differential properties of solutions of variational problems for functionals of linear growth. J Math Sci 64, 1256–1277 (1993). https://doi.org/10.1007/BF01098019
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DOI: https://doi.org/10.1007/BF01098019