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Smooth dependence on data of solutions and contact regions for a Signorini problem. (English) Zbl 1183.35150
Summary: We prove that the solutions to a 2D Poisson equation with unilateral boundary conditions of Signorini type as well as their contact intervals depend smoothly on the data. The result is based on a certain local equivalence of the unilateral boundary value problem to a smooth abstract equation in a Hilbert space and on an application of the Implicit Function Theorem to that equation.

35J86 Unilateral problems for linear elliptic equations and variational inequalities with linear elliptic operators
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
47J07 Abstract inverse mapping and implicit function theorems involving nonlinear operators
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
Full Text: DOI
[1] P. Grisvard, Behavior of the solutions of an elliptic boundary value problem in a polygonal or polyhedral domain, in: B. Hubbard (Ed.), Proceedings of the Third Symposium on the Numerical Solution of PDE, SYNSPADE 1975. pp. 207-274 · Zbl 0361.35022
[2] Mazya, V.G.; Nazarov, S.A.; Plamenevskii, B.A., Asymptotic theory of elliptic boundary value problems in singularly perturbed domains, ()
[3] Nazarov, S.A.; Plamenevskii, B.A., ()
[4] Eck, C.; Nazarov, S.A.; Wendland, W.L., Asymptotic analysis for a mixed boundary value contact problem, Arch. ration. mech. anal., 156, 4, 275-316, (2001) · Zbl 1002.74071
[5] Argatov, I.I.; Sokołowski, J., Asymptotics of the energy functional of the Signorini problem under a small singular perturbation of the domain, Comp. math. math. physics, 43, 5, 744-758, (2003) · Zbl 1074.35007
[6] Recke, L.; Eisner, J.; Kučera, M., Smooth dependence on parameters of solutions and contact regions for an obstacle problem, J. math. anal. appl., 288, 462-480, (2003) · Zbl 1042.49007
[7] Eisner, J.; Kučera, M.; Recke, L., Smooth bifurcation for an obstacle problem, Differential integral equations, 2, 121-140, (2005) · Zbl 1201.74125
[8] Eisner, J.; Kučera, M.; Recke, L., Smooth continuation of solutions and eigenvalues for variational inequalities based on the implicit function theorem, J. math. anal. appl., 274, 1, 159-180, (2002) · Zbl 1040.49006
[9] Gilbarg, D.; Trudinger, N.S., Elliptic partial differential equations of second order, (2001), Springer Berlin, Heidelberg, New York · Zbl 0691.35001
[10] Kinderlehrer, D., The smoothness of the solution of the boundary obstacle problem, J. math. pures appl., 60, 193-212, (1981) · Zbl 0459.35092
[11] Frehse, J., A regularity result for nonlinear elliptic systems, Math. Z., 121, 305-310, (1971) · Zbl 0219.35036
[12] Schumann, R., Regularity for variational inequalities—A survey of results, (), Nonconvex optim. appl., 55, 269-282, (2001) · Zbl 1043.49017
[13] Nečas, J., LES Méthodes directes en théorie des equations elliptiques, (1967), Academia Praha · Zbl 1225.35003
[14] Kufner, A.; Sändig, A.-M., Some applications of weighted Sobolev spaces, () · Zbl 0662.46034
[15] Grisvard, P., Singularities in boundary value problems, () · Zbl 0766.35001
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