Summary
We consider a (possibly) vector-valued function u: Ω→ RN, Ω⊂Rn, minimizing the integral\(\int\limits_\Omega {(\left| {D_1 u} \right|^2 + \ldots + \left| {D_{n - 1} } \right|^2 + \left| {D_n u} \right|^p )dx} \), 2-2/(n*1)<p<2, whereD i u=∂u/∂x i or some more general functional retaining the same behaviour, we prove higher integrability for Du: D1 u,..., Dn−1 u ∈ Lp/(p-1) and Dnu ∈ L2; this result allows us to get existence of second weak derivatives: D(D1 u),...,D(Dn−1u)∈L2 and D(Dn u) ∈L p.
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This work has been supported by MURST and GNAFA-CNR.
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Bhattacharya, T., Leonetti, F. On improved regularity of weak solutions of some degenerate, Anisotropic elliptic systems. Annali di Matematica pura ed applicata 170, 241–255 (1996). https://doi.org/10.1007/BF01758990
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DOI: https://doi.org/10.1007/BF01758990