Potential Analysis

, Volume 46, Issue 3, pp 403–430 | Cite as

Fractional Differentiability for Solutions of Nonlinear Elliptic Equations

  • A. L. Baisón
  • A. Clop
  • R. Giova
  • J. Orobitg
  • A. Passarelli di Napoli
Article

Abstract

We study nonlinear elliptic equations in divergence form
$$\text {div }{\mathcal A}(x,Du)=\text {div } G.$$
When \({\mathcal A}\) has linear growth in D u, and assuming that \(x\mapsto {\mathcal A}(x,\xi )\) enjoys \(B^{\alpha }_{\frac {n}\alpha , q}\) smoothness, local well-posedness is found in \(B^{\alpha }_{p,q}\) for certain values of \(p\in [2,\frac {n}{\alpha })\) and \(q\in [1,\infty ]\). In the particular case \({\mathcal A}(x,\xi )=A(x)\xi \), G = 0 and \(A\in B^{\alpha }_{\frac {n}\alpha ,q}\), \(1\leq q\leq \infty \), we obtain \(Du\in B^{\alpha }_{p,q}\) for each \(p<\frac {n}\alpha \). Our main tool in the proof is a more general result, that holds also if \({\mathcal A}\) has growth s−1 in D u, 2 ≤ sn, and asserts local well-posedness in L q for each q > s, provided that \(x\mapsto {\mathcal A}(x,\xi )\) satisfies a locally uniform VMO condition.

Keywords

Nonlinear elliptic equations Local well-posedness Higher order fractional differentiability Besov spaces 

Mathematics Subject Classification (2010)

35B65 35J60 42B37 49N60 

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Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Departament de MatemàtiquesUniversitat Autònoma de BarcelonaBellaterraCatalonia
  2. 2.Dipartimento di Studi Economici e GiuridiciUniversità degli Studi di Napoli “Parthenope”NapoliItaly
  3. 3.Dipartimento di Matematica e Appl. “R.Caccioppoli”Università degli Studi di Napoli “Federico II”NapoliItaly

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