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Remarks on certain singular perturbations with ill-posed limit in shell theory and elasticity
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 |
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 $A_x.$
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. |
show all references
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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