Smooth bifurcation for a Signorini problem on a rectangle

Jan Eisner; Milan Kučera; Lutz Recke

Mathematica Bohemica (2012)

  • Volume: 137, Issue: 2, page 131-138
  • ISSN: 0862-7959

Abstract

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We study a parameter depending semilinear elliptic PDE on a rectangle with Signorini boundary conditions on a part of one edge and mixed (zero Dirichlet and Neumann) boundary conditions on the rest of the boundary. We describe smooth branches of smooth nontrivial solutions bifurcating from the trivial solution branch in eigenvalues of the linearized problem. In particular, the contact sets of these nontrivial solutions are intervals which change smoothly along the branch. The main tools of the proof are first a certain local equivalence of the unilateral BVP to a system consisting of a corresponding classical BVP and of two scalar equations (which determine the ends of the contact intervals), and secondly an application of the classical Crandall-Rabinowitz type local bifurcation techniques (scaling and application of the Implicit Function Theorem) to that system.

How to cite

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Eisner, Jan, Kučera, Milan, and Recke, Lutz. "Smooth bifurcation for a Signorini problem on a rectangle." Mathematica Bohemica 137.2 (2012): 131-138. <http://eudml.org/doc/246972>.

@article{Eisner2012,
abstract = {We study a parameter depending semilinear elliptic PDE on a rectangle with Signorini boundary conditions on a part of one edge and mixed (zero Dirichlet and Neumann) boundary conditions on the rest of the boundary. We describe smooth branches of smooth nontrivial solutions bifurcating from the trivial solution branch in eigenvalues of the linearized problem. In particular, the contact sets of these nontrivial solutions are intervals which change smoothly along the branch. The main tools of the proof are first a certain local equivalence of the unilateral BVP to a system consisting of a corresponding classical BVP and of two scalar equations (which determine the ends of the contact intervals), and secondly an application of the classical Crandall-Rabinowitz type local bifurcation techniques (scaling and application of the Implicit Function Theorem) to that system.},
author = {Eisner, Jan, Kučera, Milan, Recke, Lutz},
journal = {Mathematica Bohemica},
keywords = {Signorini problem; smooth bifurcation; variational inequality; boundary obstacle; Crandall-Rabinowitz type theorem; Signorini problem; smooth bifurcation; variational inequality; boundary obstacle},
language = {eng},
number = {2},
pages = {131-138},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Smooth bifurcation for a Signorini problem on a rectangle},
url = {http://eudml.org/doc/246972},
volume = {137},
year = {2012},
}

TY - JOUR
AU - Eisner, Jan
AU - Kučera, Milan
AU - Recke, Lutz
TI - Smooth bifurcation for a Signorini problem on a rectangle
JO - Mathematica Bohemica
PY - 2012
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 137
IS - 2
SP - 131
EP - 138
AB - We study a parameter depending semilinear elliptic PDE on a rectangle with Signorini boundary conditions on a part of one edge and mixed (zero Dirichlet and Neumann) boundary conditions on the rest of the boundary. We describe smooth branches of smooth nontrivial solutions bifurcating from the trivial solution branch in eigenvalues of the linearized problem. In particular, the contact sets of these nontrivial solutions are intervals which change smoothly along the branch. The main tools of the proof are first a certain local equivalence of the unilateral BVP to a system consisting of a corresponding classical BVP and of two scalar equations (which determine the ends of the contact intervals), and secondly an application of the classical Crandall-Rabinowitz type local bifurcation techniques (scaling and application of the Implicit Function Theorem) to that system.
LA - eng
KW - Signorini problem; smooth bifurcation; variational inequality; boundary obstacle; Crandall-Rabinowitz type theorem; Signorini problem; smooth bifurcation; variational inequality; boundary obstacle
UR - http://eudml.org/doc/246972
ER -

References

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  1. Eisner, J., Kučera, M., Recke, L., 10.1016/j.na.2009.08.014, Nonlinear Anal., Theory Methods Appl. 72 (2010), 1358-1378. (2010) Zbl1183.35150MR2577537DOI10.1016/j.na.2009.08.014
  2. Eisner, J., Kučera, M., Recke, L., 10.1016/j.na.2010.10.058, Nonlinear Anal., Theory Methods Appl. 74 (2011), 1853-1877. (2011) Zbl1213.35233MR2764386DOI10.1016/j.na.2010.10.058
  3. Frehse, J., 10.1007/BF01109976, Math. Z. 121 (1971), 305-310. (1971) Zbl0219.35036MR0320518DOI10.1007/BF01109976
  4. Grisvard, P., Behavior of the solutions of an elliptic boundary value problem in a polygonal or polyhedral domain, Proceedings of the Third Symposium on the Numerical Solution of PDEs, SYNSPADE 1975 B. Hubbard Academic Press, New York (1976), 207-274. (1976) Zbl0361.35022MR0466912
  5. Gröger, K., 10.1007/BF01442860, Math. Ann. 283 (1989), 679-687. (1989) MR0990595DOI10.1007/BF01442860
  6. Kinderlehrer, D., The smoothness of the solution of the boundary obstacle problem, J. Math. Pures Appl. 60 (1981), 193-212. (1981) Zbl0459.35092MR0620584
  7. Nazarov, S. A., Plamenevskii, B. A., Elliptic Problems in Domains with Piecewise Smooth Boundaries, De Gruyter Expositions in Mathematics vol. 13, de Gruyter, Berlin (1994). (1994) Zbl0806.35001MR1283387

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