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Variational inequalities for energy functionals with nonstandard growth conditions. (English) Zbl 0987.49019

The authors study the partial regularity of minimizers of obstacle problems of the form
‘Minimize \(\int_\Omega G(\nabla u)dx\), where \(\Omega\) is a bounded Lipschitz domain of \(R^n\), \(u\) is a function from \(\Omega \) into \({R}\), \(u=0\) on \(\partial \Omega\), \(u\geq \Phi\), \(\Phi \in C^2(\overline \Omega)\), \(\Phi< 0\) on \(\partial \Omega\). The function \(G\) is of class \(C^2\) satisfying several growth conditions, in particular : \(C_1(A(|E|) -1) \leq G(E) \leq C_2(A(|E|) +1)\), \(D^2G(E)(Y,Y) \geq \lambda (1+|E|)^{-\mu}|Y|^2\), \(\lambda>0\), \(A\) is an \(N\)-function having the \(\Delta_2\)-property’.
It is proved that the problem admits a unique solution in a suitable subset of an Orlicz-Sobolev space. If \(n\geq 2\) and \(\mu<{{4}\over {n}}\) it is proved that the minimizer \(u\) belongs to \(C^{1,\alpha}(\Omega_0)\) for all \(0<\alpha<1\), where \(\Omega_0\subset \Omega\) is of full measure in \(\Omega\).

MSC:

49N60 Regularity of solutions in optimal control
49J40 Variational inequalities
35J70 Degenerate elliptic equations
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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