| Title:
|
Estimate of the Hausdorff measure of the singular set of a solution for a semi-linear elliptic equation associated with superconductivity (English) |
| Author:
|
Aramaki, Junichi |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
46 |
| Issue:
|
3 |
| Year:
|
2010 |
| Pages:
|
185-201 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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. (English) |
| Keyword:
|
singular set |
| Keyword:
|
semi-linear elliptic equation |
| Keyword:
|
Ginzburg-Landau system |
| MSC:
|
35B65 |
| MSC:
|
35J60 |
| MSC:
|
35J61 |
| MSC:
|
47F05 |
| MSC:
|
82D37 |
| MSC:
|
82D55 |
| idZBL:
|
Zbl 1240.82013 |
| idMR:
|
MR2735905 |
| . |
| Date available:
|
2010-10-22T05:35:28Z |
| Last updated:
|
2013-09-19 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/140787 |
| . |
| Reference:
|
[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 |
| Reference:
|
[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. Zbl 1112.82053, MR 2227157 |
| Reference:
|
[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 |
| Reference:
|
[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. Zbl 1195.82103, MR 2662062 |
| Reference:
|
[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. Zbl 1171.82020, MR 2522753 |
| Reference:
|
[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. Zbl 0084.30402, MR 0092067 |
| Reference:
|
[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 |
| Reference:
|
[8] Federer, H.: Geometric Measure Theory.Springer, Berlin, 1969. Zbl 0176.00801, MR 0257325 |
| Reference:
|
[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. MR 0833393, 10.1512/iumj.1986.35.35015 |
| Reference:
|
[10] Gilbarg, D., Trudinger, N. S.: Elliptic Partial Differential Equations of Second Order.Springer, New York, 1983. Zbl 0562.35001, MR 0737190 |
| Reference:
|
[11] Han, Q.: Singular sets of solutions to elliptic equations.Indiana Univ. Math. J. 43 (1994), 983–1002. Zbl 0817.35020, MR 1305956, 10.1512/iumj.1994.43.43043 |
| Reference:
|
[12] Han, Q.: Schauder estimates for elliptic operators with applications to nodal set.J. Geom. Anal. 10 (3) (2000), 455–480. MR 1794573, 10.1007/BF02921945 |
| Reference:
|
[13] Han, Q., Hardt, R., Lin, F.-G.: Geometric measure of singular sets of elliptic equations.Comm. Pure Appl. Math. 51 (1998), 1425–1443. Zbl 0940.35065, MR 1639155, 10.1002/(SICI)1097-0312(199811/12)51:11/12<1425::AID-CPA8>3.0.CO;2-3 |
| Reference:
|
[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 |
| Reference:
|
[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. Zbl 0851.58046, MR 1391630, 10.1006/jfan.1996.0056 |
| Reference:
|
[16] Helffer, B., Morame, A.: Magnetic bottles in connection with superconductivity.J. Funct. Anal. 185 (2001), 604–680. Zbl 1078.81023, MR 1856278, 10.1006/jfan.2001.3773 |
| Reference:
|
[17] Lu, K., Pan, X.-B.: Estimates of upper critical field for the Ginzburg-Landau equations of superconductivity.Physica D 127 (1999), 73–104. MR 1678383, 10.1016/S0167-2789(98)00246-2 |
| Reference:
|
[18] Lu, K., Pan, X.-B.: Surface nucleation of supeconductivity in $3$-dimension.J. Differential Equations 168 (2000), 386–452. MR 1808455, 10.1006/jdeq.2000.3892 |
| Reference:
|
[19] Mattila, P.: Geometry of Sets and Measures in Euclidean Spaces.Cambridge Univ. Press, 1995. Zbl 0819.28004, MR 1333890 |
| Reference:
|
[20] Morgan, F.: Geometric Measure Theory, A beginner’s Guide.fourth ed., Academic Press, 2009. Zbl 1179.49050, MR 2455580 |
| Reference:
|
[21] Pan, X.-B.: Landau-de Gennes model of liquid crystals and critical wave number.Comm. Math. Phys. 239 (2003), 343–382. Zbl 1056.49005, MR 1997445, 10.1007/s00220-003-0875-8 |
| Reference:
|
[22] Pan, X.-B.: Surface superconductivity in $3$-dimensions.Trans. Amer. Math. Soc. 356 (2004), 3899–3937. Zbl 1051.35090, MR 2058511, 10.1090/S0002-9947-04-03530-5 |
| Reference:
|
[23] Pan, X.-B.: Nodal sets of solutions of equations involving magnetic Schrödinger operator in three dimension.J. Math. Phys. 48 (2007), 053521. MR 2329883, 10.1063/1.2738752 |
| Reference:
|
[24] Pan, X.-B., Kwek, K. H.: On a problem related to vortex nucleation of superconductivity.J. Differential Equations 182 (2002), 141–168. Zbl 1064.35057, MR 1912073, 10.1006/jdeq.2001.4093 |
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