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Quasi-minima. (English) Zbl 0541.49008

A notion of quasi-minimum (Q-minimum) is defined for the functional of calculus of variations \(F(u;\Omega)=\int_{\Gamma}f(x,u(x),Du(x))dx,\) where \(\Omega\) is a bounded domain in \(R^ n\), \(u=(u^ 1,...,u^ N)\quad(N\geq 1)\) is a function defined on \(\Omega\) and f(x,u,p) is a Carathéodory function. Then it is shown that this notion includes among others the minima of variational integrals, the solutions of elliptic differential equations and of variational inequalities, the quasi-regular mappings.
The problem of the regularity of Q-minima is also discussed, and several results concerning the regularity for Q-minima in \(L^ p\) and \(C^{0\alpha}\)-spaces are obtained. In the case where \(N=1\) some qualitative properties for Q-minima like the weak maximum principle or the Liouville property are proved. A stability result for Q-minima with respect to \(\Gamma\)-convergence of a sequence of functionals is also given.
Reviewer: Z.Denkowski

MSC:

49J45 Methods involving semicontinuity and convergence; relaxation
49J10 Existence theories for free problems in two or more independent variables
35B65 Smoothness and regularity of solutions to PDEs
49J40 Variational inequalities
35J50 Variational methods for elliptic systems
49K10 Optimality conditions for free problems in two or more independent variables
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References:

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