zbMATH — the first resource for mathematics

Convex sets in \(W^ 1_ p(\Omega)\) and their normal cones. Applications to variational inequalities. (English. Russian original) Zbl 0719.46020
Sib. Math. J. 30, No. 5, 793-802 (1989); translation from Sib. Mat. Zh. 30, No. 5(177), 163-175 (1989).
Summary (translated from the Russian): “We consider variational inequalities with second-order elliptic operators in the form of equations with a multivalued right-hand side: -Au\(\in N({\mathcal K},u)\), where \(N({\mathcal K},v)\subset (W^ 1_ p(\Omega))'\) is the normal cone to the convex closed set \({\mathcal K}\subset W^ 1_ p(\Omega).\) We study a problem on the dependence of differential properties of the functions from \(A^{-1}{\mathcal N}\) (and, in particular, of the solution u) on the set of functionals \({\mathcal N}=\cup_{v\in {\mathcal K}}N(K,v)\) given in the problem. We show that \({\mathcal N}\subset X({\mathcal C})=\{F\in (W^ 1_ p(\Omega))':\) supp \(F\subseteq {\mathcal C}\}\) in problems with an obstacle on the closed set \({\mathcal E}\subseteq {\bar \Omega}\). We study properties of the subspace X(\({\mathcal C})\), the connection with the spaces of traces on \({\mathcal E}\) (we introduce the concept of a generalized trace on an arbitrary set of nonzero capacity), and the character of singularities of functions of the class \(A^{-1}X({\mathcal E})\) that are conditioned by the capacity and dimension of \({\mathcal E}\). For the case of the obstacle in \({\mathcal E}={\bar \Omega}\) we write down the necessary conditions for its solvability in the class \(C^ 2({\bar \Omega})\) that explain the presence of the smoothness threshold.”
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
49J40 Variational inequalities
35J85 Unilateral problems; variational inequalities (elliptic type) (MSC2000)
Full Text: DOI
[1] D. Kinderlehrer and G. Stampacchia, Introduction to Variational Inequalities [Russian translation], Mir, Moscow (1980). · Zbl 0457.35001
[2] A. Friedman, Variational Principles and Free Boundary Problems, Wiley-Interscience, New York (1982). · Zbl 0564.49002
[3] J. F. Rodriquez, Ostacle Problems in Mathematical Physics, Univ. of Lisbon, Lisbon (1987).
[4] G. M. Troianiello, Elliptic Differential Equations and Obstacle Problems, Plenum, New York (1987). · Zbl 0655.35002
[5] N. N. Ural’tseva, ?On the regularity of solutions of variational inequalities,? Usp. Mat. Nauk,42, No. 6 (258), 151-174 (1987).
[6] O. A. Ladyzhenskaya and N. N. Ural’tseva, Linear and Quasilinear Equations of Elliptic Type [in Russian], Nauka, Moscow (1973).
[7] J. L. Lions, Some Methods of Resolution of Nonlinear Boundary Value Problems [Russian translation], Mir, Moscow (1972).
[8] H. Brezis and G. Stampacchia, ?Sur la regularité de la solution d’inequalités,? Bull. Soc. Math.,96, No. 2, 153-180 (1968).
[9] T. N. Rozhkovskaya, ?On the question of regularity of solutions of variational inequalities and their applications to problems of mathematical physics,? Trudy Sem. Akad. Soboleva, IM SO AN SSSR, Novosibirsk (1983). · Zbl 0529.49003
[10] V. G. Mazya, Sobolev Spaces [in Russian], Leningrad State Univ. (1985).
[11] V. M. Goldstein and Yu. G. Reshetnyak, Introduction to the Theory of Functions with Generalized Derivatives and Quasiconformal Mappings [in Russian], Nauka, Moscow (1983).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.