×

Boundary regularity for solutions of degenerate elliptic equations. (English) Zbl 0675.35042

The author studies degenerate elliptic equations of the type \(div A(x,u,\nabla u)=B(x,u,\nabla u)\) with \(| p|^{-m}(\partial A_ i/\partial p_ j)(x,u,p)\) uniformly positive definite and bounded for some \(m>-1\). Under standard assumptions interior \(C_{1,\alpha}\)- estimates for bounded solutions have been proved by E. DiBenedetto [Nonlinear Anal., Theory Methods Appl. 7, 827-850 (1983; Zbl 0539.35027)] and P. Tolksdorf [J. Differ. Equations 51, 126-150 (1984; Zbl 0488.35017)]. The author shows the corresponding global regularity-results for both the Dirichlet - and the conormal boundary value problem.
Reviewer: M.Wiegner

MSC:

35J70 Degenerate elliptic equations
35J67 Boundary values of solutions to elliptic equations and elliptic systems
35B65 Smoothness and regularity of solutions to PDEs
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Campanato, S., Proprietà di Hölderianità di alcune classi di funzioni, Annali Scu. norm. sup. Pisa, 17, 175-188 (1963) · Zbl 0121.29201
[2] Campanato, S., Equazioni ellitiche de \(II^0\) ordine e spazi £\(^{2,λ}\), Annali Mat. pura appl., 69, 321-382 (1965) · Zbl 0145.36603
[3] Dibenedetto, E., \(C^{1+α}\) local regularity of weak solutions of degenerate elliptic equations, Nonlinear Analysis, 7, 827-850 (1983) · Zbl 0539.35027
[4] Evans, L. C., A new proof of local \(C^{1+α}\) regularity for solutions of certain degenerate elliptic P.D.E., J. diff. Eqns, 45, 356-373 (1982) · Zbl 0508.35036
[5] Giaquinta, M.; Giusti, E., Global \(C^{1+α}\)-regularity for second order quasilinear elliptic equations in divergence form, J. reine angew. Math., 351, 55-65 (1984) · Zbl 0528.35014
[6] Gilbarg, D.; Hörmander, L., Intermediate Schauder estimates, Archs ration. Mech. Analysis, 74, 297-318 (1980) · Zbl 0454.35022
[7] Gilbarg, D.; Trudinger, N. S., Elliptic Partial Differential Equations of Second Order (1983), Springer: Springer Berlin · Zbl 0691.35001
[8] Krylov, N. V., Boundedly inhomogeneous elliptic and parabolic equations in a domain, Izv. Akad. Nauk SSSR. Izv. Akad. Nauk SSSR, Math. USSR-Izv., 20, 67-98 (1984), (In Russian.) English translation · Zbl 0578.35024
[9] Ladyzhenskaya, O. A.; Ural’tseva, N. N., Linear and Quasilinear Elliptic Equations (1973), Nauka: Nauka Moscow, English translation: Academic Press, New York · Zbl 0269.35029
[10] Lewis, J. L., Regularity of the derivatives of solutions to certain degenerate elliptic equations, Indiana Univ. Math. Jl., 32, 849-858 (1983) · Zbl 0554.35048
[11] Lieberman, G. M., Interior gradient estimates for non-uniformly parabolic equations, Indiana Univ. Math. Jl., 32, 579-601 (1983) · Zbl 0491.35021
[12] Lieberman, G. M., The conormal derivative problem for elliptic equations of variational type, J. diff. Eqns, 49, 218-257 (1983) · Zbl 0476.35032
[13] Lieberman, G. M., Mixed boundary value problems for elliptic and parabolic differential equations of second order, J. math. Analysis Applic., 113, 422-440 (1986) · Zbl 0609.35021
[14] Lieberman, G. M., The Dirichlet problem for quasilinear elliptic equations with continuously differentiable boundary data, Communs partial diff. Eqns, 11, 167-229 (1986) · Zbl 0589.35036
[15] Lieberman, G. M., The first initial-boundary value problem for quasilinear second order parabolic equations, Annali Scu. norm. sup. Pisa, 13, 347-387 (1986) · Zbl 0655.35047
[16] Lieberman, G. M., Hölder continuity of the gradient of solutions of uniformly parabolic equations with conormal boundary conditions, Annali Mat. pura appl., 148, 397-398 (1987) · Zbl 0658.35051
[17] Simon, L. M., Interior gradient bounds for non-uniformly elliptic equations, Indiana Univ. Math. Jl., 25, 821-855 (1976) · Zbl 0346.35016
[18] Tolksdorf, P., Regularity for a more general class of quasilinear elliptic equations, J. diff. Eqns, 51, 126-150 (1984) · Zbl 0488.35017
[19] Trudinger, N. S., On Harnack type inequalities and their applications to quasilinear elliptic equations, Communs pure appl. Math., 20, 721-747 (1967) · Zbl 0153.42703
[20] Trudinger, N. S., On the Dirichlet problem for quasilinear uniformly elliptic equations in \(n\) variables, Archs ration. Mech. Analysis, 27, 108-119 (1967) · Zbl 0165.12502
[21] Uhlenbeck, K., Regularity for a class of nonlinear elliptic systems, Acta math., 138, 219-240 (1977) · Zbl 0372.35030
[22] Ural’tseva, N. N., Degenerate quasilinear systems, Zap. nauchno Sem. Leningrad. Otdel. Mat. Inst. Steklov (LOMI). Zap. nauchno Sem. Leningrad. Otdel. Mat. Inst. Steklov (LOMI), Sem. Math. V.A. Steklov Mat. Inst. Leningrad, 7, 83-99 (1968), (In Russian.) English translation in · Zbl 0199.42502
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.