Existence, comparison, and convergence results for a class of elliptic hemivariational inequalities. (English) Zbl 1486.35147
Summary: In this paper we study a class of elliptic boundary hemivariational inequalities which originates in the steady-state heat conduction problem with nonmonotone multivalued subdifferential boundary condition on a portion of the boundary described by the Clarke generalized gradient of a locally Lipschitz function. First, we prove a new existence result for the inequality employing the theory of pseudomonotone operators. Next, we give a result on comparison of solutions, and provide sufficient conditions that guarantee the asymptotic behavior of solution, when the heat transfer coefficient tends to infinity. Further, we show a result on the continuous dependence of solution on the internal energy and heat flux. Finally, some examples of convex and nonconvex potentials illustrate our hypotheses.
MSC:
| 35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |
| 35J65 | Nonlinear boundary value problems for linear elliptic equations |
| 35J87 | Unilateral problems for nonlinear elliptic equations and variational inequalities with nonlinear elliptic operators |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
References:
| [1] | Azzam, A.; Kreyszig, E., On solutions of elliptic equations satisfying mixed boundary conditions, SIAM J. Math. Anal., 13, 254-262 (1982) · Zbl 0485.35041 · doi:10.1137/0513018 |
| [2] | Bacuta, C.; Bramble, JH; Pasciak, JE, Using finite element tools in proving shift theorems for elliptic boundary value problems, Numer. Linear Algebra Appl., 10, 33-64 (2003) · Zbl 1071.65559 · doi:10.1002/nla.311 |
| [3] | Barbu, V., Boundary control problems with non linear state equation, SIAM J. Control Optim., 20, 125-143 (1982) · Zbl 0493.49024 · doi:10.1137/0320010 |
| [4] | Boukrouche, M.; Tarzia, DA; Hömberg, D.; Tröltzsch, F., On existence, uniqueness, and convergence of optimal control problems governed by parabolic variational inequalities, IFIP Advances in Information and Communication Technology 391, 76-84 (2013), Berlin: Springer, Berlin · Zbl 1266.49010 |
| [5] | Carl, S.; Le, VK; Motreanu, D., Nonsmooth Variational Problems and Their Inequalities (2007), New York: Springer, New York · Zbl 1109.35004 · doi:10.1007/978-0-387-46252-3 |
| [6] | Clarke, FH, Optimization and Nonsmooth Analysis (1983), New York: Wiley Interscience, New York · Zbl 0582.49001 |
| [7] | Denkowski, Z.; Migorski, S.; Papageorgiou, NS, An Introduction to Nonlinear Analysis: Theory (2003), Boston: Kluwer Academic, Boston · Zbl 1040.46001 · doi:10.1007/978-1-4419-9158-4 |
| [8] | Denkowski, Z.; Migorski, S.; Papageorgiou, NS, An Introduction to Nonlinear Analysis: Applications (2003), Boston: Kluwer Academic, Boston · Zbl 1054.47001 · doi:10.1007/978-1-4419-9156-0 |
| [9] | Duvaut, G.; Lions, JL, Les Inéquations en Mécanique et en Physique (1972), Paris: Dunod, Paris · Zbl 0298.73001 |
| [10] | Garguichevich, GG; Tarzia, DA, The steady-state two-fase Stefan problem with an internal energy and some related problems, Atti Sem. Mat. Fis. Univ. Modena, 39, 615-634 (1991) · Zbl 0752.35091 |
| [11] | Gasinski, L.; Liu, Z.; Migorski, S.; Ochal, A.; Peng, Z., Hemivariational inequality approach to evolutionary constrained problems on star-shaped sets, J. Optim. Theory Appl., 164, 514-533 (2015) · Zbl 1440.47048 · doi:10.1007/s10957-014-0587-6 |
| [12] | Grisvard, P., Elliptic Problems in Nonsmooth Domains (1985), London: Pitman, London · Zbl 0695.35060 |
| [13] | Kawohl, B., On nonlinear parabolic equations with abruptly changing nonlinear boundary conditions, Nonlinear Anal., 5, 1141-1153 (1981) · Zbl 0468.35050 · doi:10.1016/0362-546X(81)90008-0 |
| [14] | Lanzani, L.; Capagna, L.; Brown, RM, The mixed problem in \(L^p\) for some two-dimensional Lipschitz domain, Math. Ann., 342, 91-124 (2008) · Zbl 1180.35203 · doi:10.1007/s00208-008-0223-6 |
| [15] | Migorski, S.; Ochal, A., Boundary hemivariational inequality of parabolic type, Nonlinear Anal., 57, 579-596 (2004) · Zbl 1050.35043 · doi:10.1016/j.na.2004.03.004 |
| [16] | Migorski, S.; Ochal, A., A unified approach to dynamic contact problems in viscoelasticity, J. Elasticity, 83, 247-275 (2006) · Zbl 1138.74375 · doi:10.1007/s10659-005-9034-0 |
| [17] | Migorski, S.; Ochal, A.; Sofonea, M., Nonlinear Inclusions and Hemivariational Inequalities, Models and Analysis of Contact Problems (2013), New York: Springer, New York · Zbl 1262.49001 |
| [18] | Migorski, S.; Ochal, A.; Sofonea, M., A class of variational-hemivariational inequalities in reflexive Banach spaces, J. Elasticity, 127, 151-178 (2017) · Zbl 1368.47045 · doi:10.1007/s10659-016-9600-7 |
| [19] | Naniewicz, Z.; Panagiotopoulos, PD, Mathematical Theory of Hemivariational Inequalities and Applications (1995), New York: Marcel Dekker Inc., New York · Zbl 0968.49008 |
| [20] | Panagiotopoulos, PD, Nonconvex problems of semipermeable media and related topics, Z. Angew. Math. Mech., 65, 29-36 (1985) · Zbl 0583.34019 · doi:10.1002/zamm.19850650116 |
| [21] | Panagiotopoulos, PD, Inequality Problems in Mechanics and Applications (1985), Boston: Birkhäuser, Boston · Zbl 0579.73014 · doi:10.1007/978-1-4612-5152-1 |
| [22] | Panagiotopoulos, PD, Hemivariational Inequalities (1993), Berlin: Springer, Berlin · Zbl 0826.73002 · doi:10.1007/978-3-642-51677-1 |
| [23] | Rodrigues, JF, Obstacle Problems in Mathematical Physics (1987), Amsterdam: North-Holland, Amsterdam · Zbl 0606.73017 |
| [24] | Shillor, M.; Sofonea, M.; Telega, JJ, Models and Analysis of Quasistatic Contact (2004), Berlin: Springer, Berlin · Zbl 1069.74001 · doi:10.1007/b99799 |
| [25] | Sofonea, M.; Migorski, S., Variational-Hemivariational Inequalities with Applications (2018), Boca Raton: CRC Press, Boca Raton · Zbl 1384.49002 |
| [26] | Tabacman, ED; Tarzia, DA, Sufficient and/or necessary condition for the heat transfer coefficient on \(\Gamma_1\) and the heat flux on \(\Gamma_2\) to obtain a steady-state two-phase Stefan problem, J. Differ. Equ., 77, 16-37 (1989) · Zbl 0682.35109 · doi:10.1016/0022-0396(89)90155-1 |
| [27] | Tarzia, DA, Sur le problème de Stefan à deux phases, C. R. Acad. Sci. Paris Ser. A, 288, 941-944 (1979) · Zbl 0416.35001 |
| [28] | Tarzia, D.A.: Aplicación de métodos variacionales en el caso estacionario del problema de Stefan a dos fases. Math. Notae 27, 145-156 (1979-1980) · Zbl 0456.76093 |
| [29] | Tarzia, D.A.: Una familia de problemas que converge hacia el caso estacionario del problema de Stefan a dos fases. Math. Notae 27, 157-165 (1979-1980) · Zbl 0456.76091 |
| [30] | Tarzia, DA, An inequality for the constant heat flux to obtain a steady-state two-phase Stefan problem, Eng. Anal., 5, 4, 177-181 (1988) · Zbl 0758.35092 · doi:10.1016/0264-682X(88)90013-5 |
| [31] | Zeidler, E., Nonlinear Functional Analysis and Applications. II A/B (1990), New York: Springer, New York · Zbl 0684.47029 |
| [32] | Zeng, B.; Liu, Z.; Migorski, S., On convergence of solutions to variational-hemivariational inequalities, Z. Angew. Math. Phys., 69, 87, 1-20 (2018) · Zbl 06908352 |
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