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On the best Hölder exponent for two dimensional elliptic equations in divergence form

Author: Tonia Ricciardi
Journal: Proc. Amer. Math. Soc. 136 (2008), 2771-2783
MSC (2000): Primary 35J15
Published electronically: April 14, 2008
MathSciNet review: 2399041
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Abstract: We obtain an estimate for the Hölder continuity exponent for weak solutions to the following elliptic equation in divergence form:

$\displaystyle \mathrm{div}(A(x)\nabla u)=0 \qquad\mathrm{in }\Omega, $

where $ \Omega$ is a bounded open subset of $ \mathbb{R}^2$ and, for every $ x\in\Omega$, $ A(x)$ is a symmetric matrix with bounded measurable coefficients. Such an estimate ``interpolates'' between the well-known estimate of Piccinini and Spagnolo in the isotropic case $ A(x)=a(x)I$, where $ a$ is a bounded measurable function, and our previous result in the unit determinant case $ \det A\equiv1$. Furthermore, we show that our estimate is sharp. Indeed, for every $ \tau\in[0,1]$ we construct coefficient matrices $ A_\tau$ such that $ A_0$ is isotropic and $ A_1$ has unit determinant, and such that our estimate for $ A_\tau$ reduces to an equality, for every $ \tau\in[0,1]$.

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Additional Information

Tonia Ricciardi
Affiliation: Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università di Napoli Federico II, Via Cintia, 80126 Napoli, Italy

Keywords: Linear elliptic equation, measurable coefficients, H\"older regularity
Received by editor(s): November 25, 2005
Received by editor(s) in revised form: March 9, 2006
Published electronically: April 14, 2008
Additional Notes: The author was supported in part by the INdAM-GNAMPA Project Funzionali policonvessi e mappe quasiregolari and by the MIUR National Project Variational Methods and Nonlinear Differential Equations.
Communicated by: David S. Tartakoff
Article copyright: © Copyright 2008 American Mathematical Society

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