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**Convergence of solutions of bilateral problems in variable domains and related questions.**
*(English)*
Zbl 1448.49020

Summary: We discuss some results on the convergence of minimizers and minimum values of integral and more general functionals on sets of functions defined by bilateral constraints in variable domains. We consider the case of regular constraints, i.e., constraints lying in the corresponding Sobolev space, and the case where the lower constraint is zero and the upper constraint is an arbitrary nonnegative function. The first case concerns a larger class of integrands and requires the positivity almost everywhere of the difference between the upper and lower constraints. In the second case, this requirement is absent. Moreover, in the latter case, the exhaustion condition of an \(n\)-dimensional domain by a sequence of \(n\)-dimensional domains plays an important role. We give a series of results involving this condition. In particular, using the exhaustion condition, we prove a certain convergence of sets of functions defined by bilateral (generally irregular) constraints in variable domains.

### MSC:

49J45 | Methods involving semicontinuity and convergence; relaxation |

46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |

### Keywords:

integral functional; bilateral problem; minimizer; minimum value; \( \Gamma \)-convergence of functionals; strong connectedness of spaces; \( \mathcal{H}\)-convergence of sets; exhaustion condition
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\textit{A. A. Kovalevsky}, Ural Math. J. 3, No. 2, 51--66 (2017; Zbl 1448.49020)

### References:

[1] | De Giorgi E., Franzoni T., “Su un tipo di convergenza variazionale”, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8), 58:6 (1975), 842-850 · Zbl 0339.49005 |

[2] | Zhikov V. V., “Questions of convergence, duality, and averaging for functionals of the calculus of variations”, Math. USSR-Izv., 23:2 (1984), 243-276 · Zbl 0551.49012 |

[3] | Dal Maso G., An introduction to \(\Gamma \)-convergence, Birkhäuser, Boston, 1993, 352 pp. · Zbl 0816.49001 |

[4] | Zhikov V.V., “On passage to the limit in nonlinear variational problems”, Russian Acad. Sci. Sb. Math., 76:2 (1993), 427-459 · Zbl 0791.35036 |

[5] | Kovalevskii A.A., “Averaging of variable variational problems”, Dokl. Akad. Nauk Ukr. SSR., A, no. 8, 1988, 6-9 (in Russian) |

[6] | Kovalevskii A.A., “On the connectedness of subsets of Sobolev spaces and the \(\Gamma \)-convergence of functionals with varying domain of definition”, Nelinein. Granichnye Zadachi, 1 (1989), 48—54 (in Russian) |

[7] | Kovalevskii A.A., “On the \(\Gamma \)-convergence of integral functionals defined on Sobolev weakly connected spaces”, Ukrainian Math. J., 48:5 (1996), 683-698 · Zbl 0888.35121 |

[8] | Hruslov E.Ya., “The asymptotic behavior of solutions of the second boundary value problem under fragmentation of the boundary of the domain”, Math. USSR-Sb., 35:2 (1979), 266-282 · Zbl 0421.35019 |

[9] | Kovalevsky A.A., “On the convergence of solutions of variational problems with bilateral obstacles in variable domains”, Proc. Steklov Inst. Math. (Suppl.), 296, suppl. 1 (2017), 151-163 · Zbl 1369.49013 |

[10] | Kovalevsky A.A., “On the convergence of solutions to bilateral problems with the zero lower constraint and an arbitrary upper constraint in variable domains”, Nonlinear Anal., 147 (2016), 63-79 · Zbl 1350.49010 |

[11] | Vainberg M.M., Variational method and method of monotone operators in the theory of nonlinear equations, Wiley, New York, 1974, 356 pp. |

[12] | Kovalevskii A.A., “Some problems connected with the problem of averaging variational problems for functionals with a variable domain”, Current Analysis and its Applications, 1989, 62-70, Naukova dumka, Kiev |

[13] | Kovalevsky A.A., “Obstacle problems in variable domains”, Complex Var. Elliptic Equ., 56:12 (2011), 1071-1083 · Zbl 1227.49019 |

[14] | Adams R.A., Sobolev spaces, Academic Press, New York, 1975, 286 pp. · Zbl 0314.46030 |

[15] | Malý J., Ziemer W.P., Fine regularity of solutions of elliptic partial differential equations., AMS, Providence, 1997, 291 pp. · Zbl 0882.35001 |

[16] | Kovalevskii A.A., “\(G\)-convergence and homogenization of nonlinear elliptic operators in divergence form with variable domain”, Russian Acad. Sci. Izv. Math., 44:3 (1995), 431-460 |

[17] | Kovalevskij A.A., “Conditions of the \(\Gamma \)-convergence and homogenization of integral functionals with different domains of the definition”, Dokl. Akad. Nauk Ukr. SSR, 1991, no. 4, 5-8 (in Russian) |

[18] | Kovalevskii A.A., “On necessary and sufficient conditions for the \(\Gamma \)-convergence of integral functionals with different domains of definition”, Nelinein. Granichnye Zadachi, 4 (1992), 29-39 (in Russian) |

[19] | Kovalevskii A.A., Rudakova O.A., “On the \(\Gamma \)-compactness of integral functionals with degenerate integrands”, Nelinein. Granichnye Zadachi, 15 (2005), 149-145 (in Russian) |

[20] | Rudakova O.A., “On \(\Gamma \)-convergence of integral functionals defined on various weighted Sobolev spaces ”, Ukrainian Math. J., 61 (2009), 121-139 · Zbl 1224.46068 |

[21] | Kovalevsky A.A., “On \(L^1\)-functions with a very singular behaviour”, Nonlinear Anal., 85 (2013), 66—77 · Zbl 1282.26017 |

[22] | Murat F., Sur l’homogeneisation d’inequations elliptiques du 2ème ordre, relatives au convexe \(K(\psi_1,\psi_2)=\{v\in H^1_0(\Omega)\,\vert\,\psi_1\leqslant v\leqslant\psi_2\) p.p. dans \(\Omega\}\)., Laboratoire d’Analyse Numérique, no. 76013. Univ. Paris VI., 1976, 23 pp. |

[23] | Kovalevsky A.A., “Variational problems with unilateral pointwise functional constraints in variable domains”, Trudy Inst. Mat. i Mekh. UrO RAN, 23:2 (2017), 133-150 (in Russian) |

[24] | Kovalevsky A.A., Rudakova O.A., “Variational problems with pointwise constraints and degeneration in variable domains”, Differ. Equ. Appl., 1:4 (2009), 517-559 · Zbl 1184.49021 |

[25] | Gilbarg D., Trudinger N.S., Elliptic partial differential equations of second order, Springer, Berlin, 1983, 518 pp. · Zbl 0562.35001 |

[26] | Kuratowski K., Topology, v. I, Academic Press, New York, 1966, 580 pp. |

[27] | Mosco U., “Convergence of convex sets and of solutions of variational inequalities”, Adv. Math., 3:4 (1969), 510-585 · Zbl 0192.49101 |

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