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Existence of solutions of obstacle problems. (English) Zbl 0735.35067
The purpose of this important paper is to study the question concerning the existence of solutions to obstacle problems. The pointwise regularity of solutions to certain obstacle problems (variational inequalities) is investigated. The authors have shown that, under certain conditions (\(\Omega\) is a bounded nonempty open set of \(\mathbb{R}^ n\) and \(\psi\) is a real-valued function on \(\Omega\)), there exists \(u\in W^{1,\alpha}(\Omega)\), such that \[ u(x)\geq\psi(x) \tag{1} \] for \(x\) in a certain subset of \(\Omega\), \(u\) satisfies a given boundary condition and \[ \sum_{i=1}^ n\int_ \Omega a_ i(x,u(x),Du(x)){\partial\varphi \over\partial x_ i}(x)dx+\int_ \Omega b(x,u(x),Du(x))\varphi(x)dx\geq0 \tag{2} \] for all \(\varphi\) in a specified subspace of \(W_ 0^{1,\alpha}(\Omega)\) with (3) \(\varphi(x)\geq\psi(x)-u(x)\) for all \(x\) in a specified subset of \(\Omega\).
Main result: In the existence theorem an integral average procedure is used for extending the domain of Sobolev functions to quasi-all (in the sense of an appropriate capacity) \(\Omega\) and the authors require that (1) and (3) hold for quasi-all \(x\in\Omega\). This means that this work generalizes the work of those authors who require only that (1) and (3) hold for almost all \(x\in\Omega\).

35J85 Unilateral problems; variational inequalities (elliptic type) (MSC2000)
35D10 Regularity of generalized solutions of PDE (MSC2000)
Full Text: DOI
[1] Adams, D.R., Capacity and the obstacle problem, Appl. math. optim., 8, 39-57, (1981) · Zbl 0503.35039
[2] Federer, H.; Ziemer, W.P., The Lebesgue set of a function whose distribution derivatives are pth power summable, Indiana univ. math. J., 22, 139-158, (1972) · Zbl 0238.28015
[3] Frehse, J.; Mosco, U., Sur la continuite ponctuelle des solutions locales faibles du probleme d’obstacle, C.R. acad. sci Paris, ser. 1 math., 295, 571-574, (1982) · Zbl 0506.49004
[4] Gariepy, R.; Ziemer, W.P., A regularity condition at the boundary for solutions of quasilinear elliptic equations, Arch. ration mech. anal., 67, 25-39, (1977) · Zbl 0389.35023
[5] Gilbarg, D.; Trudinger, N.S., Elliptic partial differential equations of second order, Grundlehr. math. wiss., 224, (1977) · Zbl 0691.35001
[6] Heinonen, J.; Kilpeläinen, T., On the Wiener criterion and quasi-linear obstacle problems, Trans. am. math. soc., 310, 239-255, (1988) · Zbl 0711.35052
[7] Hess, P., A strongly non-linear elliptic boundary value problem, J. math. analysis applic., 43, 241-249, (1973) · Zbl 0267.35039
[8] Kilpeläinen T. & Ziemer W., Pointwise regularity of solutions to nonlinear double obstacle problems, Arkiv Mate. (to appear).
[9] Landes, R.; Mustonen, V., On pseudo-monotone operators and non-linear noncoercive variational problems on unbounded domains, Math. annln, 248, 241-246, (1980) · Zbl 0416.35072
[10] Landes, R.; Mustonen, V., Unilateral obstacle problems for strongly non-linear second order elliptic operators, Proc. symp. pure math., 45, (1986), Part 2 · Zbl 0601.35045
[11] Lindqvist, P., Regularity for the gradient of the solution to a non-linear obstacle problem with degenerate ellipticity, ()
[12] Lions, J.L., Quelques méthodes de résolution des problèmes aux limites nonlinéaires, (1969), Dunod Gautheire-Villans Paris · Zbl 0189.40603
[13] Leray, J.; Lions, J.L., Quelques résultats de višik sur LES problèmes elliptiques non linéaires par LES méthodes de minty – browder, Bull soc. math fr., 93, 97-107, (1965) · Zbl 0132.10502
[14] Ladyzhenskaya, O.A.; Ural’tseva, N.N., Linear and quasilinear elliptic equations, (1968), Academic Press New York · Zbl 0164.13002
[15] Maeda, F.-Y., A convergence property for solutions of certain quasi-linear elliptic equations, (), 547-553
[16] Maz’ja, V.G., On the continuity at a boundary point of solutions of quasilinear elliptic equations, Vestvak leningrad univ., Vestvak leningrad univ. math., 3, No. 13, (1976)
[17] Meyers, N.; Elcrat, A., Some results on regularity for solutions of nonlinear elliptic systems and quasi-regular functions, Duke math. J., 42, 121-136, (1975) · Zbl 0347.35039
[18] Meyers, N.G., A theory of capacities for potentials of functions in Lebesgue classes, Math. scand., 26, 255-292, (1970) · Zbl 0242.31006
[19] Michael, J.H., A general theory for linear elliptic partial differential equations, J. diff eqns, 23, 1-29, (1977) · Zbl 0299.35037
[20] Michael, J.H., The dirchlet problem for quasilinear uniformly elliptic equations, Nonlinear analysis, 9, 455-467, (1985) · Zbl 0526.35031
[21] Mustonen, V., A class of strongly non-linear variational inequalities in unbounded domains, J. London math. soc., 19, 319-328, (1979) · Zbl 0393.35023
[22] Michael, J.H.; Ziemer, W.P., A Lusin type approximation of Sobolev functions by smooth functions, Contemp. math., 42, 135-167, (1985) · Zbl 0592.41031
[23] Michael, J.H.; Ziemer, W.P., Interior regularity for solutions to obstacle problems, Nonlinear analysis, 10, 1427-1448, (1986) · Zbl 0603.49006
[24] Michael, J.H.; Ziemer, W.P., The Wiener criterion and quasilinear uniformly elliptic equations, Ann. inst. Henri Poincaré, 4, 453-486, (1987) · Zbl 0635.35030
[25] Saks S., Theory of the Integral. Hafner, New York.
[26] Stein, E., Singular intervals and differentiability properties of functions, (1970), Princeton University Press New Jersey
[27] Trudinger, N.S., On harnark type inequalities and their application to quasilinear elliptic equations, Communs pure appl. math., 20, 721-747, (1987) · Zbl 0153.42703
[28] Wiener, N., The Dirichlet problem, J. math. phys., 3, 127-146, (1924) · JFM 51.0361.01
[29] Wiener, N., Certain notions in potential theory, J. math. phys., 3, 24-51, (1924) · JFM 51.0360.05
[30] Yosida, K., Functional analysis, (1974), Springer New York · Zbl 0152.32102
[31] Ziemer, W.P., Boundary regularity for quasiminima, Archs ration mech. analysis, 92, 371-382, (1986) · Zbl 0611.35030
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