# American Institute of Mathematical Sciences

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

 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.
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
 [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.
 [1] Luigi C. Berselli, Carlo R. Grisanti. On the regularity up to the boundary for certain nonlinear elliptic systems. Discrete and Continuous Dynamical Systems - S, 2016, 9 (1) : 53-71. doi: 10.3934/dcdss.2016.9.53 [2] Hugo Beirão da Veiga. Elliptic boundary value problems in spaces of continuous functions. Discrete and Continuous Dynamical Systems - S, 2016, 9 (1) : 43-52. doi: 10.3934/dcdss.2016.9.43 [3] Santiago Cano-Casanova. Coercivity of elliptic mixed boundary value problems in annulus of $\mathbb{R}^N$. Discrete and Continuous Dynamical Systems, 2012, 32 (11) : 3819-3839. doi: 10.3934/dcds.2012.32.3819 [4] Mark I. Vishik, Sergey Zelik. Attractors for the nonlinear elliptic boundary value problems and their parabolic singular limit. Communications on Pure and Applied Analysis, 2014, 13 (5) : 2059-2093. doi: 10.3934/cpaa.2014.13.2059 [5] Matthias Eller, Daniel Toundykov. Carleman estimates for elliptic boundary value problems with applications to the stablization of hyperbolic systems. Evolution Equations and Control Theory, 2012, 1 (2) : 271-296. doi: 10.3934/eect.2012.1.271 [6] Shujie Li, Zhitao Zhang. Multiple solutions theorems for semilinear elliptic boundary value problems with resonance at infinity. Discrete and Continuous Dynamical Systems, 1999, 5 (3) : 489-493. doi: 10.3934/dcds.1999.5.489 [7] Zongming Guo, Yunting Yu. Boundary value problems for a semilinear elliptic equation with singular nonlinearity. Communications on Pure and Applied Analysis, 2016, 15 (2) : 399-412. doi: 10.3934/cpaa.2016.15.399 [8] Angelo Favini, Rabah Labbas, Stéphane Maingot, Maëlis Meisner. Boundary value problem for elliptic differential equations in non-commutative cases. Discrete and Continuous Dynamical Systems, 2013, 33 (11&12) : 4967-4990. doi: 10.3934/dcds.2013.33.4967 [9] Huyuan Chen, Hichem Hajaiej, Ying Wang. Boundary blow-up solutions to fractional elliptic equations in a measure framework. Discrete and Continuous Dynamical Systems, 2016, 36 (4) : 1881-1903. doi: 10.3934/dcds.2016.36.1881 [10] Claudia Anedda, Giovanni Porru. Second order estimates for boundary blow-up solutions of elliptic equations. Conference Publications, 2007, 2007 (Special) : 54-63. doi: 10.3934/proc.2007.2007.54 [11] Colin J. Cotter, Darryl D. Holm. Geodesic boundary value problems with symmetry. Journal of Geometric Mechanics, 2010, 2 (1) : 51-68. doi: 10.3934/jgm.2010.2.51 [12] Paolo Piersanti. On the improved interior regularity of a boundary value problem modelling the displacement of a linearly elastic elliptic membrane shell subject to an obstacle. Discrete and Continuous Dynamical Systems, 2022, 42 (2) : 1011-1037. doi: 10.3934/dcds.2021145 [13] Hiroshi Watanabe. Solvability of boundary value problems for strongly degenerate parabolic equations with discontinuous coefficients. Discrete and Continuous Dynamical Systems - S, 2014, 7 (1) : 177-189. doi: 10.3934/dcdss.2014.7.177 [14] Iñigo U. Erneta. Well-posedness for boundary value problems for coagulation-fragmentation equations. Kinetic and Related Models, 2020, 13 (4) : 815-835. doi: 10.3934/krm.2020028 [15] Oleksandr Boichuk, Victor Feruk. Boundary-value problems for weakly singular integral equations. Discrete and Continuous Dynamical Systems - B, 2022, 27 (3) : 1379-1395. doi: 10.3934/dcdsb.2021094 [16] Olha P. Kupenko, Rosanna Manzo. On optimal controls in coefficients for ill-posed non-Linear elliptic Dirichlet boundary value problems. Discrete and Continuous Dynamical Systems - B, 2018, 23 (4) : 1363-1393. doi: 10.3934/dcdsb.2018155 [17] Youngmok Jeon, Dongwook Shin. Immersed hybrid difference methods for elliptic boundary value problems by artificial interface conditions. Electronic Research Archive, 2021, 29 (5) : 3361-3382. doi: 10.3934/era.2021043 [18] Zhijun Zhang. Boundary blow-up for elliptic problems involving exponential nonlinearities with nonlinear gradient terms and singular weights. Communications on Pure and Applied Analysis, 2007, 6 (2) : 521-529. doi: 10.3934/cpaa.2007.6.521 [19] Daniela De Silva, Fausto Ferrari, Sandro Salsa. On two phase free boundary problems governed by elliptic equations with distributed sources. Discrete and Continuous Dynamical Systems - S, 2014, 7 (4) : 673-693. doi: 10.3934/dcdss.2014.7.673 [20] Laurence Halpern, Jeffrey Rauch. Hyperbolic boundary value problems with trihedral corners. Discrete and Continuous Dynamical Systems, 2016, 36 (8) : 4403-4450. doi: 10.3934/dcds.2016.36.4403

2020 Impact Factor: 1.392