Choe, Hi Jun A regularity theory for a general class of quasilinear elliptic partial differential equations and obstacle problems. (English) Zbl 0733.35024 Arch. Ration. Mech. Anal. 114, No. 4, 383-394 (1991). Let \(u\in W^{1,p}(\Omega)\) be a solution of the variational inequality \[ \int_{\Omega}| Du|^{p-2} Du(Dv-Du)dx\geq \int_{\Omega}b(x,u,Du)(v-u)dx+\int_{\Omega}f(x)(Dv-Du)dx \] for all \(v\in \{W_ 0^{1,p}(\Omega)+u_ 0\}\), v(x)\(\geq \psi (x)\) a.e. in \(\Omega\), where \(u_ 0\in W^{1,p}(\Omega)\) and \(u_ 0(x)\geq \psi (x)\). Under appropriate conditions on b and f the author shows that \(u\in C_{loc}^{0,\alpha}(\Omega)\) when \(\psi =\in W_{loc}^{1,m}(\Omega)\) \((m>n)\) and \(u\in C_{loc}^{1,\alpha}(\Omega)\) when \(\psi \in W_{loc}^{1,\beta}(\Omega)\) \((\alpha >0)\) for some suitable \(\beta >0\). Reviewer: G.Porru (Cagliari) Cited in 1 ReviewCited in 55 Documents MSC: 35D10 Regularity of generalized solutions of PDE (MSC2000) 35J85 Unilateral problems; variational inequalities (elliptic type) (MSC2000) Keywords:Hölder continuity; obstacle problems; variational inequality PDFBibTeX XMLCite \textit{H. J. Choe}, Arch. Ration. Mech. Anal. 114, No. 4, 383--394 (1991; Zbl 0733.35024) Full Text: DOI References: [1] H. J. Choe & J. Lewis, On the obstacle problem for quasilinear elliptic equation of p Laplacian type, to appear in SIAM J. Math. Analysis. · Zbl 0762.35035 [2] E. DiBenedetto, C 1,? local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal. 7 (1983), 827-850. · Zbl 0539.35027 · doi:10.1016/0362-546X(83)90061-5 [3] M. Fuchs, Hölder continuity of the gradient for degenerate variational inequalities, Bonn Lecture Notes, 1989. · Zbl 0701.49019 [4] M. Giaquinta, Multiple integrals in the Calculus of Variations and Non-linear Elliptic Systems, Annals of Math. Studies, Vol. 105, Princeton University Press, 1983. · Zbl 0516.49003 [5] J. Lewis, Regularity of derivatives of solutions to certain degenerate elliptic equations, Indiana Univ. Math. J. 32 (1983), 849-858. · Zbl 0554.35048 · doi:10.1512/iumj.1983.32.32058 [6] G. Lieberman, The natural generalization of the natural conditions of Ladyzhenskaya and Ural’tseva for elliptic equations, to appear in Comm. Partial Diff. Eqs. · Zbl 0742.35028 [7] P. Lindquist, Regularity for the gradient of the solution to a nonlinear obstacle problem with degenerate ellipticity, Nonlinear Anal. 12 (1988), 1245-1255. · doi:10.1016/0362-546X(88)90056-9 [8] O. A. Ladyzhenskaya & N. N. Ural’tseva, Linear and Quasilinear Elliptic Equations, Academic Press, 1968. [9] J. Manfredi, Regularity for minima of functionals with p-growth, J. Diff. Eqs. 76 (1988), 203-212. · Zbl 0674.35008 · doi:10.1016/0022-0396(88)90070-8 [10] J. Michael & W. Ziemer, Interior regularity for solutions to obstacle problems, Nonlinear Anal. 10 (1986), 1427-1448. · Zbl 0603.49006 · doi:10.1016/0362-546X(86)90113-6 [11] T. Norando, C 1,? local regularity for a class of quasilinear elliptic variational inequalities, Boll. Un. Ital. Mat. 5 (1986), 281-291. · Zbl 0639.49009 [12] J. Serrin, Local behavior of solutions of quasi-linear elliptic equations, Acta Math. 111 (1964), 247-302. · Zbl 0128.09101 · doi:10.1007/BF02391014 [13] P. Tolksdorff, Regularity for a more general class of quasi-linear elliptic equations, J. Diff. Eqs. 51 (1984), 126-150. · Zbl 0522.35018 · doi:10.1016/0022-0396(84)90105-0 [14] N. Trudinger, On Harnack type inequalities and their application to quasilinear elliptic equations, Comm. Pure Appl. Math. 20 (1967), 721-747. · Zbl 0153.42703 · doi:10.1002/cpa.3160200406 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.