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Keywords:
Signorini problem; smooth bifurcation; variational inequality; boundary obstacle; Crandall-Rabinowitz type theorem
Signorini problem; smooth bifurcation; variational inequality; boundary obstacle; Crandall-Rabinowitz type theorem
Summary:
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.
References:
[1] Eisner, J., Kučera, M., Recke, L.: Smooth dependence on data of solutions and contact regions for a Signorini problem. Nonlinear Anal., Theory Methods Appl. 72 (2010), 1358-1378. DOI 10.1016/j.na.2009.08.014 | MR 2577537 | Zbl 1183.35150
[2] Eisner, J., Kučera, M., Recke, L.: Smooth bifurcation branches of solutions for a Signorini problem. Nonlinear Anal., Theory Methods Appl. 74 (2011), 1853-1877. DOI 10.1016/j.na.2010.10.058 | MR 2764386 | Zbl 1213.35233
[3] Frehse, J.: A regularity result for nonlinear elliptic systems. Math. Z. 121 (1971), 305-310. DOI 10.1007/BF01109976 | MR 0320518 | Zbl 0219.35036
[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. MR 0466912 | Zbl 0361.35022
[5] Gröger, K.: A $W^{1,p}$-estimate for solutions to mixed boundary value problems for second order elliptic differential equations. Math. Ann. 283 (1989), 679-687. DOI 10.1007/BF01442860 | MR 0990595
[6] Kinderlehrer, D.: The smoothness of the solution of the boundary obstacle problem. J. Math. Pures Appl. 60 (1981), 193-212. MR 0620584 | Zbl 0459.35092
[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). MR 1283387 | Zbl 0806.35001
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