Fractional Differentiability for Solutions of Nonlinear Elliptic Equations

• Published in: Potential analysis 2017-03, Vol.46 (3), p.403-430
• Main Authors: Baisón, A. L., Clop, A., Giova, R., Orobitg, J., Passarelli di Napoli, A.
• Format: Article
• Language: English
• Subjects: Divergence; Functional Analysis; Geometry; Mathematical analysis; Mathematics; Mathematics and Statistics; Nonlinear equations; Potential Theory; Probability Theory and Stochastic Processes; Smoothness; Well posed problems
• ISSN: 0926-2601; 1572-929X
• DOI: 10.1007/s11118-016-9585-7

We study nonlinear elliptic equations in divergence form div A ( x , Du ) = div G . When A has linear growth in D u , and assuming that x ↦ A ( x , ξ ) enjoys B n α , q α smoothness, local well-posedness is found in B p , q α for certain values of p ∈ [ 2 , n α ) and q ∈ [ 1 , ∞ ] . In the particular case A ( x , ξ ) = A ( x ) ξ , G = 0 and A ∈ B n α , q α , 1 ≤ q ≤ ∞ , we obtain Du ∈ B p , q α for each p < n α . Our main tool in the proof is a more general result, that holds also if A has growth s −1 in D u , 2 ≤ s ≤ n , and asserts local well-posedness in L q for each q > s , provided that x ↦ A ( x , ξ ) satisfies a locally uniform VMO condition.