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Keywords:
singular set; semi-linear elliptic equation; Ginzburg-Landau system
singular set; semi-linear elliptic equation; Ginzburg-Landau system
Summary:
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.
References:
[1] Aramaki, J.: On an elliptic model with general nonlinearity associated with superconductivity. Int. J. Differ. Equ. Appl. 10 (4) (2006), 449–466. MR 2321824
[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 2227157 | Zbl 1112.82053
[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 2478221
[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 2662062 | Zbl 1195.82103
[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 2522753 | Zbl 1171.82020
[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 0092067 | Zbl 0084.30402
[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 1309733
[8] Federer, H.: Geometric Measure Theory. Springer, Berlin, 1969. MR 0257325 | Zbl 0176.00801
[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 10.1512/iumj.1986.35.35015 | MR 0833393
[10] Gilbarg, D., Trudinger, N. S.: Elliptic Partial Differential Equations of Second Order. Springer, New York, 1983. MR 0737190 | Zbl 0562.35001
[11] Han, Q.: Singular sets of solutions to elliptic equations. Indiana Univ. Math. J. 43 (1994), 983–1002. DOI 10.1512/iumj.1994.43.43043 | MR 1305956 | Zbl 0817.35020
[12] Han, Q.: Schauder estimates for elliptic operators with applications to nodal set. J. Geom. Anal. 10 (3) (2000), 455–480. DOI 10.1007/BF02921945 | MR 1794573
[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 10.1002/(SICI)1097-0312(199811/12)51:11/12<1425::AID-CPA8>3.0.CO;2-3 | MR 1639155 | Zbl 0940.35065
[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 1728303
[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 10.1006/jfan.1996.0056 | MR 1391630 | Zbl 0851.58046
[16] Helffer, B., Morame, A.: Magnetic bottles in connection with superconductivity. J. Funct. Anal. 185 (2001), 604–680. DOI 10.1006/jfan.2001.3773 | MR 1856278 | Zbl 1078.81023
[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 10.1016/S0167-2789(98)00246-2 | MR 1678383
[18] Lu, K., Pan, X.-B.: Surface nucleation of supeconductivity in $3$-dimension. J. Differential Equations 168 (2000), 386–452. DOI 10.1006/jdeq.2000.3892 | MR 1808455
[19] Mattila, P.: Geometry of Sets and Measures in Euclidean Spaces. Cambridge Univ. Press, 1995. MR 1333890 | Zbl 0819.28004
[20] Morgan, F.: Geometric Measure Theory, A beginner’s Guide. fourth ed., Academic Press, 2009. MR 2455580 | Zbl 1179.49050
[21] Pan, X.-B.: Landau-de Gennes model of liquid crystals and critical wave number. Comm. Math. Phys. 239 (2003), 343–382. DOI 10.1007/s00220-003-0875-8 | MR 1997445 | Zbl 1056.49005
[22] Pan, X.-B.: Surface superconductivity in $3$-dimensions. Trans. Amer. Math. Soc. 356 (2004), 3899–3937. DOI 10.1090/S0002-9947-04-03530-5 | MR 2058511 | Zbl 1051.35090
[23] Pan, X.-B.: Nodal sets of solutions of equations involving magnetic Schrödinger operator in three dimension. J. Math. Phys. 48 (2007), 053521. DOI 10.1063/1.2738752 | MR 2329883
[24] Pan, X.-B., Kwek, K. H.: On a problem related to vortex nucleation of superconductivity. J. Differential Equations 182 (2002), 141–168. DOI 10.1006/jdeq.2001.4093 | MR 1912073 | Zbl 1064.35057
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