American Institute of Mathematical Sciences

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

Regularity in Campanato spaces for solutions of fully nonlinear elliptic systems

 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 $$\left\{\begin{array}{l} u\in H^2\cap H^1_0(\Omega,\mathbb{R}^N) \\ F(x,D^2 u(x))= f(x), \quad \text{a.e. in} \,\,\,\Omega, \end{array} \right.$$ 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 $A_x.$
Citation: Luisa Fattorusso, Antonio Tarsia. Regularity in Campanato spaces for solutions of fully nonlinear elliptic systems. Discrete and Continuous Dynamical Systems, 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, American Mathematical Society, Providence, RI, 1995. [2] S. Campanato, Equazioni ellittiche non variazionali a coefficienti continui, Ann. Mat. Pura Appl. (4), 86 (1970), 125-154. [3] S. Campanato, A Cordes type condition for nonlinear non-variational systems, Rend. Accad. Naz. Sci XL, Mem. Mat., 13 (1989), 307-321. [4] S. Campanato, Nonvariational basic elliptic systems of second order, Rend. Sem. Fis. Milano, 60 (1990), 113-131 (1993). [5] L. Fattorusso and A. Tarsia, Morrey regularity of solutions of fully non-linear elliptic systems, Complex Var. Elliptic Equ., 55 (2010), 537-548. doi: 10.1080/17476930802657624. [6] F. W Gehring, The $L^p-$integrability of the partial derivatives of a quasiconformal mapping, Acta Math., 130 (1973), 265-277. doi: 10.1007/BF02392268. [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), 145-169. [8] E. Giusti, "Equazioni Ellittiche del Secondo Ordine," Pitagora editrice, Bologna, 1978. [9] S. Fu\vcík, O. John and A. Kufner, "Function Spaces," Monographs and Textbooks on Mechanics of Solids and Fluids, Mechanics: Analysis, Noordhoff International Publishing, Leyden, Academia, Prague, 1977. [10] M. Marino and A. Maugeri, Boundary regularity results for non-variational basic elliptic systems. in "Partial Differential Equations" (Pisa, 2000), Matematiche (Catania), 55 (2000), suppl. 2, 109-123, (2001). [11] A. Maugeri, D. K. Palagachev and L. G. Softova, "Elliptic and Parabolic Equations with Discontinuous Coefficients," Mathematical Res., 109, Wiley-VCH, 2002. [12] A. Tarsia, On Cordes and Campanato condition, Arch. Inequal. Appl., 2 (2004), 25-39. [13] A. Tarsia, Near operators theory and fully nonlinear elliptic equations, J. Global Optim., 40 (2008), 443-453. doi: 10.1007/s10898-007-9227-0.

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References:
 [1] X. Cabré and L. A. Caffarelli, "Fully Nonlinear Elliptic Equations," American Mathematical Society Colloquium Publications, 43, American Mathematical Society, Providence, RI, 1995. [2] S. Campanato, Equazioni ellittiche non variazionali a coefficienti continui, Ann. Mat. Pura Appl. (4), 86 (1970), 125-154. [3] S. Campanato, A Cordes type condition for nonlinear non-variational systems, Rend. Accad. Naz. Sci XL, Mem. Mat., 13 (1989), 307-321. [4] S. Campanato, Nonvariational basic elliptic systems of second order, Rend. Sem. Fis. Milano, 60 (1990), 113-131 (1993). [5] L. Fattorusso and A. Tarsia, Morrey regularity of solutions of fully non-linear elliptic systems, Complex Var. Elliptic Equ., 55 (2010), 537-548. doi: 10.1080/17476930802657624. [6] F. W Gehring, The $L^p-$integrability of the partial derivatives of a quasiconformal mapping, Acta Math., 130 (1973), 265-277. doi: 10.1007/BF02392268. [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), 145-169. [8] E. Giusti, "Equazioni Ellittiche del Secondo Ordine," Pitagora editrice, Bologna, 1978. [9] S. Fu\vcík, O. John and A. Kufner, "Function Spaces," Monographs and Textbooks on Mechanics of Solids and Fluids, Mechanics: Analysis, Noordhoff International Publishing, Leyden, Academia, Prague, 1977. [10] M. Marino and A. Maugeri, Boundary regularity results for non-variational basic elliptic systems. in "Partial Differential Equations" (Pisa, 2000), Matematiche (Catania), 55 (2000), suppl. 2, 109-123, (2001). [11] A. Maugeri, D. K. Palagachev and L. G. Softova, "Elliptic and Parabolic Equations with Discontinuous Coefficients," Mathematical Res., 109, Wiley-VCH, 2002. [12] A. Tarsia, On Cordes and Campanato condition, Arch. Inequal. Appl., 2 (2004), 25-39. [13] A. Tarsia, Near operators theory and fully nonlinear elliptic equations, J. Global Optim., 40 (2008), 443-453. doi: 10.1007/s10898-007-9227-0.
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