no. 1
Qualitative analysis of rupture solutions for a MEMS problem
Annales de l'I.H.P. Analyse non linéaire, Tome 33 (2016) no. 1, pp. 221-242.

We prove sharp Hölder continuity and an estimate of rupture sets for sequences of solutions of the following nonlinear problem with negative exponent

Δu=1upinΩ,p>1.
As a consequence, we prove the existence of rupture solutions with isolated ruptures in a bounded convex domain in R2.

DOI : 10.1016/j.anihpc.2014.09.009
Mots clés : Semilinear elliptic equations with negative power, Hölder continuity, Monotonicity formula 
@article{AIHPC_2016__33_1_221_0,
     author = {D\'avila, Juan and Wang, Kelei and Wei, Juncheng},
     title = {Qualitative analysis of rupture solutions for a {MEMS} problem},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {221--242},
     publisher = {Elsevier},
     volume = {33},
     number = {1},
     year = {2016},
     doi = {10.1016/j.anihpc.2014.09.009},
     mrnumber = {3436432},
     zbl = {1350.35093},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1016/j.anihpc.2014.09.009/}
}
TY  - JOUR
AU  - Dávila, Juan
AU  - Wang, Kelei
AU  - Wei, Juncheng
TI  - Qualitative analysis of rupture solutions for a MEMS problem
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2016
SP  - 221
EP  - 242
VL  - 33
IS  - 1
PB  - Elsevier
UR  - http://archive.numdam.org/articles/10.1016/j.anihpc.2014.09.009/
DO  - 10.1016/j.anihpc.2014.09.009
LA  - en
ID  - AIHPC_2016__33_1_221_0
ER  - 
%0 Journal Article
%A Dávila, Juan
%A Wang, Kelei
%A Wei, Juncheng
%T Qualitative analysis of rupture solutions for a MEMS problem
%J Annales de l'I.H.P. Analyse non linéaire
%D 2016
%P 221-242
%V 33
%N 1
%I Elsevier
%U http://archive.numdam.org/articles/10.1016/j.anihpc.2014.09.009/
%R 10.1016/j.anihpc.2014.09.009
%G en
%F AIHPC_2016__33_1_221_0
Dávila, Juan; Wang, Kelei; Wei, Juncheng. Qualitative analysis of rupture solutions for a MEMS problem. Annales de l'I.H.P. Analyse non linéaire, Tome 33 (2016) no. 1, pp. 221-242. doi : 10.1016/j.anihpc.2014.09.009. http://archive.numdam.org/articles/10.1016/j.anihpc.2014.09.009/

[1] Bethuel, F. On the singular set of stationary harmonic maps, Manuscr. Math., Volume 78 (1993) no. 4, pp. 417–443 | MR | Zbl

[2] Caffarelli, L. The obstacle problem revisited, J. Fourier Anal. Appl., Volume 4 (1998) no. 4–5, pp. 383–402 | MR | Zbl

[3] Caffarelli, L.A.; Lin, F.H. Singularly perturbed elliptic systems and multi-valued harmonic functions with free boundaries, J. Am. Math. Soc., Volume 21 (2008), pp. 847–862 | DOI | MR | Zbl

[4] Dávila, J.; Ponce, A.C. Hausdorff dimension of ruptures sets and removable singularities, C. R. Math. Acad. Sci. Paris, Volume 346 (2008) no. 1–2, pp. 27–32 | MR | Zbl

[5] Dupaigne, L.; Ponce, A.C.; Porretta, A. Elliptic equations with vertical asymptotes in the nonlinear term, J. Anal. Math., Volume 98 (2006), pp. 349–396 | DOI | MR | Zbl

[6] Esposito, P.; Ghoussoub, N.; Guo, Y. Compactness along the branch of semistable and unstable solutions for an elliptic problem with a singular nonlinearity, Commun. Pure Appl. Math., Volume 60 (2007), pp. 1731–1768 | DOI | MR | Zbl

[7] Esposito, P.; Ghoussoub, N.; Guo, Y. Mathematical Analysis of Partial Differential Equations Modeling Electrostatic MEMS, Courant Lect. Notes Math., vol. 20, Courant Institute of Mathematical Sciences/American Mathematical Society, New York/Providence, RI, 2010 | DOI | MR | Zbl

[8] Evans, L.C. Partial regularity for stationary harmonic maps into spheres, Arch. Ration. Mech. Anal., Volume 116 (1991) no. 2, pp. 101–113 | DOI | MR | Zbl

[9] Garofalo, N.; Lin, F.H. Monotonicity properties of variational integrals, Ap weights and unique continuation, Indiana Univ. Math. J., Volume 35 (1986) no. 2, pp. 245–268 | DOI | MR | Zbl

[10] Giusti, E. Minimal Surfaces and Functions of Bounded Variation, Springer Monogr. Math., vol. 80, Birkhäuser, 1984 | MR | Zbl

[11] Guo, Z.; Wei, J. Hausdorff dimension of ruptures for solutions of a semilinear elliptic equation with singular nonlinearity, Manuscr. Math., Volume 120 (2006) no. 2, pp. 193–209 | MR | Zbl

[12] Guo, Z.; Wei, J. Asymptotic behavior of touch-down solutions and global bifurcations for an elliptic problem with a singular nonlinearity, Commun. Pure Appl. Anal., Volume 7 (2008) no. 4, pp. 765–786 | MR | Zbl

[13] Jiang, H.; Lin, F.-H. Zero set of Sobolev functions with negative power of integrability, Chin. Ann. Math., Ser. B, Volume 25 (2004) no. 1, pp. 65–72 | DOI | MR | Zbl

[14] Ma, L.; Wei, J. Properties of positive solutions to an elliptic equation with negative exponent, J. Funct. Anal., Volume 254 (2008) no. 4, pp. 1058–1087 | MR | Zbl

[15] Noris, B.; Tavares, H.; Terracini, S.; Verzini, G. Uniform Hölder bounds for nonlinear Schrödinger systems with strong competition, Commun. Pure Appl. Math., Volume 63 (2010) no. 3, pp. 267–302 | DOI | MR | Zbl

[16] Pacard, F. Partial regularity for weak solutions of a nonlinear elliptic equation, Manuscr. Math., Volume 79 (1993) no. 2, pp. 161–172 | MR | Zbl

[17] Phillips, D. A minimization problem and the regularity of solutions in the presence of a free boundary, Indiana Univ. Math. J., Volume 32 (1983) no. 1, pp. 1–17 | DOI | MR | Zbl

[18] Pelesko, J.A.; Bernstein, D.H. Modeling MEMS and NEMS, Chapman & Hall/CRC, Boca Raton, FL, 2003 | MR | Zbl

[19] Rabinowitz, P.H. Some global results for nonlinear eigenvalue problems, J. Funct. Anal., Volume 7 (1971), pp. 487–513 | DOI | MR | Zbl

[20] Simon, L. Lectures on Geometric Measure Theory, Proceedings of the Centre for Mathematical Analysis, Australian National University, vol. 3, Australian National University, Centre for Mathematical Analysis, Canberra, 1983 | MR | Zbl

Cité par Sources :