Higher differentiability results for solutions to a class of non-homogeneous elliptic problems under sub-quadratic growth conditions

We prove a sharp higher differentiability result for local minimizers of functionals of the form ℱ ( w , Ω ) = ∫ Ω [ F ( x , D w ( x ) ) − f ( x ) ⋅ w ( x ) ] d x with non-autonomous integrand F ( x , ξ ) which is convex with respect to the gradient variable, under p -growth conditions, with 1 <...

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Bibliographic Details
Published in:Bulletin of mathematical sciences 2023-08, Vol.13 (2)
Main Authors: Clop, Albert, Gentile, Andrea, Passarelli di Napoli, Antonia
Format: Article
Language:English
Subjects:
a priori bounded minimizers
Convex functionals
higher differentiability
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Summary:We prove a sharp higher differentiability result for local minimizers of functionals of the form ℱ ( w , Ω ) = ∫ Ω [ F ( x , D w ( x ) ) − f ( x ) ⋅ w ( x ) ] d x with non-autonomous integrand F ( x , ξ ) which is convex with respect to the gradient variable, under p -growth conditions, with 1 < p < 2 . The main novelty here is that the results are obtained assuming that the partial map x ↦ D ξ F ( x , ξ ) has weak derivatives in some Lebesgue space L q and the datum f is assumed to belong to a suitable Lebesgue space L r . We also prove that it is possible to weaken the assumption on the datum f and on the map x ↦ D ξ F ( x , ξ ) , if the minimizers are assumed to be a priori bounded.
ISSN:1664-3607
1664-3615
DOI:10.1142/S166436072350008X