Geodesic
Estimate of the Hausdorff measure of the singular set of a solution for a semi-linear elliptic equation associated with superconductivity
Archivum mathematicum, Tome 46 (2010) no. 3, pp. 185-201

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We study the boundedness of the Hausdorff measure of the singular set of any solution for a semi-linear elliptic equation in general dimensional Euclidean space ${\mathbb{R}}^n$. In our previous paper, we have clarified the structures of the nodal set and singular set of a solution for the semi-linear elliptic equation. In particular, we showed that the singular set is $(n-2)$-rectifiable. In this paper, we shall show that under some additive smoothness assumptions, the $(n-2) $-dimensional Hausdorff measure of singular set of any solution is locally finite.
We study the boundedness of the Hausdorff measure of the singular set of any solution for a semi-linear elliptic equation in general dimensional Euclidean space ${\mathbb{R}}^n$. In our previous paper, we have clarified the structures of the nodal set and singular set of a solution for the semi-linear elliptic equation. In particular, we showed that the singular set is $(n-2)$-rectifiable. In this paper, we shall show that under some additive smoothness assumptions, the $(n-2) $-dimensional Hausdorff measure of singular set of any solution is locally finite.
Classification : 35B65, 35J60, 35J61, 47F05, 82D37, 82D55
Keywords: singular set; semi-linear elliptic equation; Ginzburg-Landau system 
Aramaki, Junichi. Estimate of the Hausdorff measure of the singular set of a solution for a semi-linear elliptic equation associated with superconductivity. Archivum mathematicum, Tome 46 (2010) no. 3, pp. 185-201. http://geodesic.mathdoc.fr/item/ARM_2010_46_3_a2/
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     title = {Estimate of the {Hausdorff} measure of the singular set of a solution for a semi-linear elliptic equation associated with superconductivity},
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     year = {2010},
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     zbl = {1240.82013},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ARM_2010_46_3_a2/}
}
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