American Institute of Mathematical Sciences

Advanced
December  2011, 31(4): 1307-1323. doi: 10.3934/dcds.2011.31.1307

Regularity in Campanato spaces for solutions of fully nonlinear elliptic systems

Luisa Fattorusso 1, and  Antonio Tarsia 2,

1. 

Dipartimento di Informatica, Matematica, Elettronica e Trasporti, Università degli Studi Mediterranea di Reggio Calabria, Loc. Feo di Vito, I-89060 Reggio Calabria, Italy
2. 

Dipartimento di Matematica “L. Tonelli”, Università di Pisa, Largo B. Pontecorvo, 5. I-56127 Pisa, Italy

Received  November 2009 Revised  March 2010 Published  September 2011

Let $\Omega$ be a bounded convex open set of $\mathbb{R}^n,$ $n\geq 2,$ $\partial \Omega $ of class $C^{2,1}.$ We consider the following Dirichlet problem \begin{equation} \left\{\begin{array}{l} u\in H^2\cap H^1_0(\Omega,\mathbb{R}^N) \\ F(x,D^2 u(x))= f(x), \quad \text{a.e. in} \,\,\,\Omega, \end{array} \right. \end{equation} where $f\in {\mathcal L}^{2,\lambda}(\Omega,\mathbb{R}^N),$ $n \leq$ $\lambda< n+2$, $F$ satisfies Campanato's Condition $A_x$ and is Hölder continuous in $\Omega$ with exponent $b.$
        We show that there exist $\varepsilon, \overline{\varepsilon}\in (0,1),$ ($\varepsilon,\overline{\varepsilon}$ depend on $\gamma$ and $\delta$), such that for any $\zeta \in (0,\overline{\varepsilon}\, n) ,$ and $ \mu \in( 0,\lambda],$ with $ \mu< (2b+\zeta)\wedge [\epsilon\,(n+2)],$ we have $D^2 u \in {\mathcal L}^{2,\mu}(\Omega,\mathbb{R}^{n^2N}),$ where $\varepsilon$ and $\overline{\varepsilon}$ depend on the constants appearing in Condition $A_x.$
Keywords: Fully nonlinear elliptic systems, Campanato spaces..
Mathematics Subject Classification: Primary: 35J55, 35B65; Secondary: 35J5.
Citation: Luisa Fattorusso, Antonio Tarsia. Regularity in Campanato spaces for solutions of fully nonlinear elliptic systems. Discrete & Continuous Dynamical Systems - A, 2011, 31 (4) : 1307-1323. doi: 10.3934/dcds.2011.31.1307
References:
[1]

X. Cabré and L. A. Caffarelli, "Fully Nonlinear Elliptic Equations,", American Mathematical Society Colloquium Publications, 43 (1995).   Google Scholar

[2]

S. Campanato, Equazioni ellittiche non variazionali a coefficienti continui,, Ann. Mat. Pura Appl. (4), 86 (1970), 125.   Google Scholar

[3]

S. Campanato, A Cordes type condition for nonlinear non-variational systems,, Rend. Accad. Naz. Sci XL, 13 (1989), 307.   Google Scholar

[4]

S. Campanato, Nonvariational basic elliptic systems of second order,, Rend. Sem. Fis. Milano, 60 (1990), 113.   Google Scholar

[5]

L. Fattorusso and A. Tarsia, Morrey regularity of solutions of fully non-linear elliptic systems,, Complex Var. Elliptic Equ., 55 (2010), 537.  doi: 10.1080/17476930802657624.  Google Scholar

[6]

F. W Gehring, The $L^p-$integrability of the partial derivatives of a quasiconformal mapping,, Acta Math., 130 (1973), 265.  doi: 10.1007/BF02392268.  Google Scholar

[7]

M. Giaquinta and G. Modica, Regularity results for some classes of highter order non linear elliptic systems,, J. Reine Angew. Math., 311/312 (1979), 145.   Google Scholar

[8]

E. Giusti, "Equazioni Ellittiche del Secondo Ordine,", Pitagora editrice, (1978).   Google Scholar

[9]

S. Fu\vcík, O. John and A. Kufner, "Function Spaces,", Monographs and Textbooks on Mechanics of Solids and Fluids, (1977).   Google Scholar

[10]

M. Marino and A. Maugeri, Boundary regularity results for non-variational basic elliptic systems., in, 55 (2000), 109.   Google Scholar

[11]

A. Maugeri, D. K. Palagachev and L. G. Softova, "Elliptic and Parabolic Equations with Discontinuous Coefficients,", Mathematical Res., 109 (2002).   Google Scholar

[12]

A. Tarsia, On Cordes and Campanato condition,, Arch. Inequal. Appl., 2 (2004), 25.   Google Scholar

[13]

A. Tarsia, Near operators theory and fully nonlinear elliptic equations,, J. Global Optim., 40 (2008), 443.  doi: 10.1007/s10898-007-9227-0.  Google Scholar

show all references

References:
[1]

X. Cabré and L. A. Caffarelli, "Fully Nonlinear Elliptic Equations,", American Mathematical Society Colloquium Publications, 43 (1995).   Google Scholar

[2]

S. Campanato, Equazioni ellittiche non variazionali a coefficienti continui,, Ann. Mat. Pura Appl. (4), 86 (1970), 125.   Google Scholar

[3]

S. Campanato, A Cordes type condition for nonlinear non-variational systems,, Rend. Accad. Naz. Sci XL, 13 (1989), 307.   Google Scholar

[4]

S. Campanato, Nonvariational basic elliptic systems of second order,, Rend. Sem. Fis. Milano, 60 (1990), 113.   Google Scholar

[5]

L. Fattorusso and A. Tarsia, Morrey regularity of solutions of fully non-linear elliptic systems,, Complex Var. Elliptic Equ., 55 (2010), 537.  doi: 10.1080/17476930802657624.  Google Scholar

[6]

F. W Gehring, The $L^p-$integrability of the partial derivatives of a quasiconformal mapping,, Acta Math., 130 (1973), 265.  doi: 10.1007/BF02392268.  Google Scholar

[7]

M. Giaquinta and G. Modica, Regularity results for some classes of highter order non linear elliptic systems,, J. Reine Angew. Math., 311/312 (1979), 145.   Google Scholar

[8]

E. Giusti, "Equazioni Ellittiche del Secondo Ordine,", Pitagora editrice, (1978).   Google Scholar

[9]

S. Fu\vcík, O. John and A. Kufner, "Function Spaces,", Monographs and Textbooks on Mechanics of Solids and Fluids, (1977).   Google Scholar

[10]

M. Marino and A. Maugeri, Boundary regularity results for non-variational basic elliptic systems., in, 55 (2000), 109.   Google Scholar

[11]

A. Maugeri, D. K. Palagachev and L. G. Softova, "Elliptic and Parabolic Equations with Discontinuous Coefficients,", Mathematical Res., 109 (2002).   Google Scholar

[12]

A. Tarsia, On Cordes and Campanato condition,, Arch. Inequal. Appl., 2 (2004), 25.   Google Scholar

[13]

A. Tarsia, Near operators theory and fully nonlinear elliptic equations,, J. Global Optim., 40 (2008), 443.  doi: 10.1007/s10898-007-9227-0.  Google Scholar

[1]

Pierpaolo Soravia. Uniqueness results for fully nonlinear degenerate elliptic equations with discontinuous coefficients. Communications on Pure & Applied Analysis, 2006, 5 (1) : 213-240. doi: 10.3934/cpaa.2006.5.213
[2]

Limei Dai. Entire solutions with asymptotic behavior of fully nonlinear uniformly elliptic equations. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1707-1714. doi: 10.3934/cpaa.2011.10.1707
[3]

Hongxia Zhang, Ying Wang. Liouville results for fully nonlinear integral elliptic equations in exterior domains. Communications on Pure & Applied Analysis, 2018, 17 (1) : 85-112. doi: 10.3934/cpaa.2018006
[4]

Fabio Camilli, Claudio Marchi. On the convergence rate in multiscale homogenization of fully nonlinear elliptic problems. Networks & Heterogeneous Media, 2011, 6 (1) : 61-75. doi: 10.3934/nhm.2011.6.61
[5]

Chunhui Qiu, Rirong Yuan. On the Dirichlet problem for fully nonlinear elliptic equations on annuli of metric cones. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5707-5730. doi: 10.3934/dcds.2017247
[6]

Robert Jensen, Andrzej Świech. Uniqueness and existence of maximal and minimal solutions of fully nonlinear elliptic PDE. Communications on Pure & Applied Analysis, 2005, 4 (1) : 199-207. doi: 10.3934/cpaa.2005.4.187
[7]

Martino Bardi, Paola Mannucci. On the Dirichlet problem for non-totally degenerate fully nonlinear elliptic equations. Communications on Pure & Applied Analysis, 2006, 5 (4) : 709-731. doi: 10.3934/cpaa.2006.5.709
[8]

Antonio Vitolo, Maria E. Amendola, Giulio Galise. On the uniqueness of blow-up solutions of fully nonlinear elliptic equations. Conference Publications, 2013, 2013 (special) : 771-780. doi: 10.3934/proc.2013.2013.771
[9]

Agnid Banerjee. A note on the unique continuation property for fully nonlinear elliptic equations. Communications on Pure & Applied Analysis, 2015, 14 (2) : 623-626. doi: 10.3934/cpaa.2015.14.623
[10]

Pablo Blanc. A lower bound for the principal eigenvalue of fully nonlinear elliptic operators. Communications on Pure & Applied Analysis, 2020, 19 (7) : 3613-3623. doi: 10.3934/cpaa.2020158
[11]

Shigeaki Koike, Andrzej Świech. Local maximum principle for $L^p$-viscosity solutions of fully nonlinear elliptic PDEs with unbounded coefficients. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1897-1910. doi: 10.3934/cpaa.2012.11.1897
[12]

Paola Mannucci. The Dirichlet problem for fully nonlinear elliptic equations non-degenerate in a fixed direction. Communications on Pure & Applied Analysis, 2014, 13 (1) : 119-133. doi: 10.3934/cpaa.2014.13.119
[13]

Fabio Punzo. Phragmèn-Lindelöf principles for fully nonlinear elliptic equations with unbounded coefficients. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1439-1461. doi: 10.3934/cpaa.2010.9.1439
[14]

Luca Rossi. Non-existence of positive solutions of fully nonlinear elliptic equations in unbounded domains. Communications on Pure & Applied Analysis, 2008, 7 (1) : 125-141. doi: 10.3934/cpaa.2008.7.125
[15]

Italo Capuzzo Dolcetta, Antonio Vitolo. Glaeser's type gradient estimates for non-negative solutions of fully nonlinear elliptic equations. Discrete & Continuous Dynamical Systems - A, 2010, 28 (2) : 539-557. doi: 10.3934/dcds.2010.28.539
[16]

N. V. Chemetov. Nonlinear hyperbolic-elliptic systems in the bounded domain. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1079-1096. doi: 10.3934/cpaa.2011.10.1079
[17]

Shuhong Chen, Zhong Tan. Optimal interior partial regularity for nonlinear elliptic systems. Discrete & Continuous Dynamical Systems - A, 2010, 27 (3) : 981-993. doi: 10.3934/dcds.2010.27.981
[18]

Luigi C. Berselli, Carlo R. Grisanti. On the regularity up to the boundary for certain nonlinear elliptic systems. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 53-71. doi: 10.3934/dcdss.2016.9.53
[19]

Yuri Latushkin, Jan Prüss, Ronald Schnaubelt. Center manifolds and dynamics near equilibria of quasilinear parabolic systems with fully nonlinear boundary conditions. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 595-633. doi: 10.3934/dcdsb.2008.9.595
[20]

Nobuyuki Kato, Norio Kikuchi. Campanato-type boundary estimates for Rothe's scheme to parabolic partial differential systems with constant coefficients. Discrete & Continuous Dynamical Systems - A, 2007, 19 (4) : 737-760. doi: 10.3934/dcds.2007.19.737

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (23)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences