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December  2011, 31(4): 1347-1363. doi: 10.3934/dcds.2011.31.1347

On linear and nonlinear elliptic boundary value problems in the plane with discontinuous coefficients

Sofia Giuffrè 1, and  Giovanna Idone 1,

1. 

D.I.M.E.T., Department of Computer Science, Mathematics, Electronics and Transportations, “Mediterranea” University of Reggio Calabria, 89060 Reggio Calabria, Italy, Italy

Received  March 2010 Revised  September 2010 Published  September 2011

Global Hölder regularity of the gradient in Morrey spaces is established for planar elliptic discontinuous equations, estimating in an explicit way the Hölder exponent in terms of the eigenvalues of the operator. The result is proved for Dirichlet or normal derivative problems and for nonlinear operators.
Keywords: boundary value problems., elliptic equations, Regularity up to the boundary.
Mathematics Subject Classification: Primary: 35J25; Secondary: 35J6.
Citation: Sofia Giuffrè, Giovanna Idone. On linear and nonlinear elliptic boundary value problems in the plane with discontinuous coefficients. Discrete & Continuous Dynamical Systems, 2011, 31 (4) : 1347-1363. doi: 10.3934/dcds.2011.31.1347
References:
[1]

L. Bers and L. Nirenberg, "On Linear and Non-Linear Elliptic Boundary Value Problems in the Plane," Atti del Convegno Internazionale sulle Equazioni lineari alle Derivate Parziali, Trieste, (1954), 141-167.  Google Scholar

[2]

S. Campanato, Un risultato relativo ad equazioni ellittiche del secondo ordine di tipo non variazionale, Ann. Scuola Norm. Sup. Pisa (3), 21 (1967), 701-707.  Google Scholar

[3]

S. Campanato, "Quaderni del Dottorato," Scuola Normale Superiore, Pisa, 1980. Google Scholar

[4]

S. Campanato, A maximum principle for nonlinear elliptic systems: Boundary fundamental estimates, Adv. Math., 66 (1987), 291-317. doi: 10.1016/0001-8708(87)90037-5.  Google Scholar

[5]

S. Campanato, Equazioni ellittiche del II ordine espazi $\mathcalL$$^(2, \lambda)$, Ann. Mat. Pura Appl., 69 (1965), 321-381. doi: 10.1007/BF02414377.  Google Scholar

[6]

E. De Giorgi, Sulla differenziabilità e l'analiticità delle estremali degli integrali multipli regolari, Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat. (3), 3 (1957), 25-43.  Google Scholar

[7]

R. Finn and J. Serrin, On the Hölder continuity of quasi-conformal and elliptic mappings, Trans. Amer. Math. Soc., 89 (1958), 1-15.  Google Scholar

[8]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Reprint of the 1998 edition, Classics in Mathematics, Springer-Verlag, Berlin, 2001.  Google Scholar

[9]

S. Giuffrè, Oblique derivative problem for nonlinear elliptic discontinuous operators in the plane with quadratic growth, C. R. Acad. Sci. Paris Sér. I, Math., 328 (1999), 859-864.  Google Scholar

[10]

S. Giuffrè, Global Hölder regularity for discontinuous elliptic equations in the plane, Proc. Amer. Math. Soc., 132 (2004), 1333-1344. doi: 10.1090/S0002-9939-03-07348-9.  Google Scholar

[11]

S. Giuffrè, Strong solvability of boundary value contact problems, Appl. Math. Optimization, 51 (2005), 361-372. doi: 10.1007/s00245-004-0817-7.  Google Scholar

[12]

S. Giuffrè and G. Idone, Global regularity for solutions to Dirichlet problem for discontinuous elliptic systems with nonlinearity $q>1$ and with natural growth, Journal of Global Optimization, 40 (2008), 99-117.  Google Scholar

[13]

E. Giusti, Sulla regolarità delle soluzioni di una classe di equazioni ellittiche, Rend. Semin. Mat. Univ. Padova, 39 (1967), 362-375.  Google Scholar

[14]

P. Hartman, Hölder continuity and non-linear elliptic partial differential equations, Duke Math. J., 25 (1957), 57-65. doi: 10.1215/S0012-7094-58-02506-7.  Google Scholar

[15]

A. Maugeri, D. K. Palagachev and L. Softova, "Elliptic and Parabolic Equations with Discontinuous Coefficients," Mathematical Research, 109, Wiley, VCH Verlag Berlin GmbH, Berlin, 2000.  Google Scholar

[16]

C. B. Morrey Jr., On the solutions of quasi-linear elliptic partial differential equations, Trans. Amer. Math. Soc., 43 (1938), 126-166. doi: 10.1090/S0002-9947-1938-1501936-8.  Google Scholar

[17]

L. Nirenberg, On nonlinear elliptic partial differential equations and Hölder continuity, Comm. Pure Appl. Math., 6 (1953), 103-156. doi: 10.1002/cpa.3160060105.  Google Scholar

[18]

D. K. Palagachev, Global strong solvability of Dirichlet problem for a class of nonlinear elliptic equations in the plane, Matematiche (Catania), 48 (1993), 311-321.  Google Scholar

[19]

L. Softova, An integral estimate for the gradient for a class of nonlinear elliptic equations in the plane, Z. Anal. Anwen, 17 (1998), 57-66.  Google Scholar

[20]

G. Talenti, Equazioni ellittiche in due variabili, Matematiche (Catania), 21 (1966), 339-376.  Google Scholar

[21]

N. S. Trudinger, "Nonlinear Second Order Elliptic Equations," Lecture Notes of Math. Inst. of Nankai Univ., Tianjin, 1986. Google Scholar

show all references

References:
[1]

L. Bers and L. Nirenberg, "On Linear and Non-Linear Elliptic Boundary Value Problems in the Plane," Atti del Convegno Internazionale sulle Equazioni lineari alle Derivate Parziali, Trieste, (1954), 141-167.  Google Scholar

[2]

S. Campanato, Un risultato relativo ad equazioni ellittiche del secondo ordine di tipo non variazionale, Ann. Scuola Norm. Sup. Pisa (3), 21 (1967), 701-707.  Google Scholar

[3]

S. Campanato, "Quaderni del Dottorato," Scuola Normale Superiore, Pisa, 1980. Google Scholar

[4]

S. Campanato, A maximum principle for nonlinear elliptic systems: Boundary fundamental estimates, Adv. Math., 66 (1987), 291-317. doi: 10.1016/0001-8708(87)90037-5.  Google Scholar

[5]

S. Campanato, Equazioni ellittiche del II ordine espazi $\mathcalL$$^(2, \lambda)$, Ann. Mat. Pura Appl., 69 (1965), 321-381. doi: 10.1007/BF02414377.  Google Scholar

[6]

E. De Giorgi, Sulla differenziabilità e l'analiticità delle estremali degli integrali multipli regolari, Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat. (3), 3 (1957), 25-43.  Google Scholar

[7]

R. Finn and J. Serrin, On the Hölder continuity of quasi-conformal and elliptic mappings, Trans. Amer. Math. Soc., 89 (1958), 1-15.  Google Scholar

[8]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Reprint of the 1998 edition, Classics in Mathematics, Springer-Verlag, Berlin, 2001.  Google Scholar

[9]

S. Giuffrè, Oblique derivative problem for nonlinear elliptic discontinuous operators in the plane with quadratic growth, C. R. Acad. Sci. Paris Sér. I, Math., 328 (1999), 859-864.  Google Scholar

[10]

S. Giuffrè, Global Hölder regularity for discontinuous elliptic equations in the plane, Proc. Amer. Math. Soc., 132 (2004), 1333-1344. doi: 10.1090/S0002-9939-03-07348-9.  Google Scholar

[11]

S. Giuffrè, Strong solvability of boundary value contact problems, Appl. Math. Optimization, 51 (2005), 361-372. doi: 10.1007/s00245-004-0817-7.  Google Scholar

[12]

S. Giuffrè and G. Idone, Global regularity for solutions to Dirichlet problem for discontinuous elliptic systems with nonlinearity $q>1$ and with natural growth, Journal of Global Optimization, 40 (2008), 99-117.  Google Scholar

[13]

E. Giusti, Sulla regolarità delle soluzioni di una classe di equazioni ellittiche, Rend. Semin. Mat. Univ. Padova, 39 (1967), 362-375.  Google Scholar

[14]

P. Hartman, Hölder continuity and non-linear elliptic partial differential equations, Duke Math. J., 25 (1957), 57-65. doi: 10.1215/S0012-7094-58-02506-7.  Google Scholar

[15]

A. Maugeri, D. K. Palagachev and L. Softova, "Elliptic and Parabolic Equations with Discontinuous Coefficients," Mathematical Research, 109, Wiley, VCH Verlag Berlin GmbH, Berlin, 2000.  Google Scholar

[16]

C. B. Morrey Jr., On the solutions of quasi-linear elliptic partial differential equations, Trans. Amer. Math. Soc., 43 (1938), 126-166. doi: 10.1090/S0002-9947-1938-1501936-8.  Google Scholar

[17]

L. Nirenberg, On nonlinear elliptic partial differential equations and Hölder continuity, Comm. Pure Appl. Math., 6 (1953), 103-156. doi: 10.1002/cpa.3160060105.  Google Scholar

[18]

D. K. Palagachev, Global strong solvability of Dirichlet problem for a class of nonlinear elliptic equations in the plane, Matematiche (Catania), 48 (1993), 311-321.  Google Scholar

[19]

L. Softova, An integral estimate for the gradient for a class of nonlinear elliptic equations in the plane, Z. Anal. Anwen, 17 (1998), 57-66.  Google Scholar

[20]

G. Talenti, Equazioni ellittiche in due variabili, Matematiche (Catania), 21 (1966), 339-376.  Google Scholar

[21]

N. S. Trudinger, "Nonlinear Second Order Elliptic Equations," Lecture Notes of Math. Inst. of Nankai Univ., Tianjin, 1986. Google Scholar

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