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The singular set of the minima of certain quadratic functionals. (English) Zbl 0543.49018

The authors study the regularity of local minima \(\bar u\in W^{1,2}_{loc}(\Omega,{\mathbb{R}}^ n)\) of the integral functional \[ \int_{B}A_{ij}^{\alpha \beta}(x,u)D_{\alpha}u^ iD_{\beta}u^ idx \] with \(\Omega =\Omega^ 0\subset {\mathbb{R}}^ n\) and B the unit ball. It is well known, that \(\bar u\) admits a set of singularities which is not empty in general [see M. Miranda and the second author, Boll. Unione Mat. Ital., IV. Ser. 1, 219-226 (1968; Zbl 0155.445)]. However, a previous result of the authors [Acta Math. 148, 31-46 (1982; Zbl 0494.49031)] shows that every \(\bar u\) is Hölder continuous in an open subset \(\Omega_ 0\) and moreover, the singular set \(\Sigma =\Omega -\Omega_ 0\) has Hausdorff dimension less than n-2. In this paper they continue and improve such analysis in the case of coefficients of the form \(A_{ij}^{\alpha \beta}(x,u)=g_{ij}(x,u)G^{\alpha \beta}(x),\) with \(G^{\alpha \beta}=G^{\beta \alpha}.\) They prove that, if \(\bar u\) is bounded, then \(\Sigma\) has Hausdorff dimension not greater than n-3 and moreover, if \(n=3\), it consists at most of isolated points.
Reviewer: A.Salvadori

MSC:

49Q20 Variational problems in a geometric measure-theoretic setting
49J45 Methods involving semicontinuity and convergence; relaxation
26B35 Special properties of functions of several variables, Hölder conditions, etc.
35D10 Regularity of generalized solutions of PDE (MSC2000)
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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References:

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