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Some properties of extremals of the functional of potential energy. (English. Russian original) Zbl 1453.49018
J. Math. Sci., New York 252, No. 2, 225-231 (2021); translation from Itogi Nauki Tekh., Ser. Sovrem. Mat. Prilozh., Temat. Obz. 152, 103-109 (2018).
Summary: In this paper, we discuss stability and instability criteria for extremal surfaces of a special functional, which is a linear combination of an area-type functional and the functional of volumetric force density. Extremals of such functionals can serve as models of physically equilibrium tents or liquids in potential gravitational fields, so that the problem of their stability or instability is very topical. Our results are based on various geometric properties of surfaces; they are obtained by methods developed by V. M. Miklyukov and V. A. Klyachin.
MSC:
49Q10 Optimization of shapes other than minimal surfaces
53A07 Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces
31A15 Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions
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