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

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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

Abstract
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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 and 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.
[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.
[3]	S. Campanato, "Quaderni del Dottorato," Scuola Normale Superiore, Pisa, 1980.
[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.
[5]	S. Campanato, Equazioni ellittiche del II ordine espazi $\mathcalL$$^(2, \lambda)$, Ann. Mat. Pura Appl., 69 (1965), 321-381. doi: 10.1007/BF02414377.
[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.
[7]	R. Finn and J. Serrin, On the Hölder continuity of quasi-conformal and elliptic mappings, Trans. Amer. Math. Soc., 89 (1958), 1-15.
[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.
[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.
[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.
[11]	S. Giuffrè, Strong solvability of boundary value contact problems, Appl. Math. Optimization, 51 (2005), 361-372. doi: 10.1007/s00245-004-0817-7.
[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.
[13]	E. Giusti, Sulla regolarità delle soluzioni di una classe di equazioni ellittiche, Rend. Semin. Mat. Univ. Padova, 39 (1967), 362-375.
[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.
[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.
[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.
[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.
[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.
[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.
[20]	G. Talenti, Equazioni ellittiche in due variabili, Matematiche (Catania), 21 (1966), 339-376.
[21]	N. S. Trudinger, "Nonlinear Second Order Elliptic Equations," Lecture Notes of Math. Inst. of Nankai Univ., Tianjin, 1986.

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.
[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.
[3]	S. Campanato, "Quaderni del Dottorato," Scuola Normale Superiore, Pisa, 1980.
[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.
[5]	S. Campanato, Equazioni ellittiche del II ordine espazi $\mathcalL$$^(2, \lambda)$, Ann. Mat. Pura Appl., 69 (1965), 321-381. doi: 10.1007/BF02414377.
[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.
[7]	R. Finn and J. Serrin, On the Hölder continuity of quasi-conformal and elliptic mappings, Trans. Amer. Math. Soc., 89 (1958), 1-15.
[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.
[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.
[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.
[11]	S. Giuffrè, Strong solvability of boundary value contact problems, Appl. Math. Optimization, 51 (2005), 361-372. doi: 10.1007/s00245-004-0817-7.
[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.
[13]	E. Giusti, Sulla regolarità delle soluzioni di una classe di equazioni ellittiche, Rend. Semin. Mat. Univ. Padova, 39 (1967), 362-375.
[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.
[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.
[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.
[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.
[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.
[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.
[20]	G. Talenti, Equazioni ellittiche in due variabili, Matematiche (Catania), 21 (1966), 339-376.
[21]	N. S. Trudinger, "Nonlinear Second Order Elliptic Equations," Lecture Notes of Math. Inst. of Nankai Univ., Tianjin, 1986.

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