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.”

##### MSC:

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

49J40 | Variational inequalities |

35J85 | Unilateral problems; variational inequalities (elliptic type) (MSC2000) |

##### Keywords:

variational inequalities with second-order elliptic operators; multivalued right-hand side; normal cone; spaces of traces; character of singularities of functions
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\textit{T. N. Rozhkovskaya}, Sib. Math. J. 30, No. 5, 793--802 (1989; Zbl 0719.46020); translation from Sib. Mat. Zh. 30, No. 5(177), 163--175 (1989)

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##### References:

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