Regularity in Campanato spaces for solutions of fully nonlinearelliptic systems

Luisa Fattorusso; Antonio Tarsia

doi:10.3934/dcds.2011.31.1307

Article Contents

2011, Volume 31, Issue 4: 1307-1323. Doi: 10.3934/dcds.2011.31.1307

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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
2.
Dipartimento di Matematica “L. Tonelli”, Università di Pisa, Largo B. Pontecorvo, 5. I-56127 Pisa

Received: October 31, 2009

Revised: February 28, 2010

Published: September 2011

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Abstract

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: 35J57.

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References

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