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 
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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/

[1] Aramaki, J.: On an elliptic model with general nonlinearity associated with superconductivity. Int. J. Differ. Equ. Appl. 10 (4) (2006), 449–466. | MR

[2] Aramaki, J.: On an elliptic problem with general nonlinearity associated with superheating field in the theory of superconductivity. Int. J. Pure Appl. Math. 28 (1) (2006), 125–148. | MR | Zbl

[3] Aramaki, J.: A remark on a semi-linear elliptic problem with the de Gennes boundary condition associated with superconductivity. Int. J. Pure Appl. Math. 50 (1) (2008), 97–110. | MR

[4] Aramaki, J.: Nodal sets and singular sets of solutions for semi-linear elliptic equations associated with superconductivity. Far East J. Math. Sci. 38 (2) (2010), 137–179. | MR | Zbl

[5] Aramaki, J., Nurmuhammad, A., Tomioka, S.: A note on a semi-linear elliptic problem with the de Gennes boundary condition associated with superconductivity. Far East J. Math. Sci. 32 (2) (2009), 153–167. | MR | Zbl

[6] Aronszajn, N.: A unique continuation theorem for solutions of elliptic partial differential equations or inequalities of second order. J. Math. Pures Appl. (9) 36 (1957), 235–249. | MR | Zbl

[7] Elliot, C. M., Matano, H., Tang, Q.: Zeros of a complex Ginzburg-Landau order parameter with applications to superconductivity. European J. Appl. Math. 5 (1994), 431–448. | MR

[8] Federer, H.: Geometric Measure Theory. Springer, Berlin, 1969. | MR | Zbl

[9] Garofalo, N., Lin, F.-H.: Monotonicity properties of variational integrals, $A_p$ weights and unique continuation. Indiana Univ. Math. J. 35 (2) (1986), 245–268. | DOI | MR

[10] Gilbarg, D., Trudinger, N. S.: Elliptic Partial Differential Equations of Second Order. Springer, New York, 1983. | MR | Zbl

[11] Han, Q.: Singular sets of solutions to elliptic equations. Indiana Univ. Math. J. 43 (1994), 983–1002. | DOI | MR | Zbl

[12] Han, Q.: Schauder estimates for elliptic operators with applications to nodal set. J. Geom. Anal. 10 (3) (2000), 455–480. | DOI | MR

[13] Han, Q., Hardt, R., Lin, F.-G.: Geometric measure of singular sets of elliptic equations. Comm. Pure Appl. Math. 51 (1998), 1425–1443. | DOI | MR | Zbl

[14] Hardt, R., Hoffmann-Ostenhof, M., Hoffmann-Ostenhof, T., Nadirashivili, N.: Critical sets of solutions to elliptic equations. J. Differential Geom. 51 (1999), 359–373. | MR

[15] Helffer, B., Mohamed, A.: Semiclassical analysis for the ground state energy of a Schrödinger operator with magnetic wells. J. Funct. Anal. 138 (1996), 40–81. | DOI | MR | Zbl

[16] Helffer, B., Morame, A.: Magnetic bottles in connection with superconductivity. J. Funct. Anal. 185 (2001), 604–680. | DOI | MR | Zbl

[17] Lu, K., Pan, X.-B.: Estimates of upper critical field for the Ginzburg-Landau equations of superconductivity. Physica D 127 (1999), 73–104. | DOI | MR

[18] Lu, K., Pan, X.-B.: Surface nucleation of supeconductivity in $3$-dimension. J. Differential Equations 168 (2000), 386–452. | DOI | MR

[19] Mattila, P.: Geometry of Sets and Measures in Euclidean Spaces. Cambridge Univ. Press, 1995. | MR | Zbl

[20] Morgan, F.: Geometric Measure Theory, A beginner’s Guide. fourth ed., Academic Press, 2009. | MR | Zbl

[21] Pan, X.-B.: Landau-de Gennes model of liquid crystals and critical wave number. Comm. Math. Phys. 239 (2003), 343–382. | DOI | MR | Zbl

[22] Pan, X.-B.: Surface superconductivity in $3$-dimensions. Trans. Amer. Math. Soc. 356 (2004), 3899–3937. | DOI | MR | Zbl

[23] Pan, X.-B.: Nodal sets of solutions of equations involving magnetic Schrödinger operator in three dimension. J. Math. Phys. 48 (2007), 053521. | DOI | MR

[24] Pan, X.-B., Kwek, K. H.: On a problem related to vortex nucleation of superconductivity. J. Differential Equations 182 (2002), 141–168. | DOI | MR | Zbl