Abstract
In this paper we prove a conjecture by P.-L. Lions on maximal regularity of L^q-type for periodic solutions to -Delta u + |Du|^gamma = f in mathbb {R}^d, under the (sharp) assumption that q > d frac{gamma -1}{gamma }.
Highlights
We address here the so-called problem of maximal Lq -regularity for equations of the form
Lions in a series of seminars and lectures (e.g. [31,32]), where he conjectured its general validity under the assumption that q
Such a condition has been improved in the finer scale of Lorentz-Morrey spaces, and end-point situations typically require additional smallness assumptions [19,23]
Summary
It has been observed that due to the superlinear nature of the problem, its (weak) solvability requires f ∈ Lq , where q Such a condition has been improved in the finer scale of Lorentz-Morrey spaces, and end-point situations typically require additional smallness assumptions [19,23]. By the fact that k → {|Du| k} |Du| − k is continuous and vanishes as k → ∞, the γq second case can be ruled out, and boundedness of {|Du| k} |Du| − k can be recovered up to k = 0 This second key step has been inspired by an interesting argument that appeared in [20] (see [21]), where W 1,2 estimates of (powers of) u are obtained arguing on superlevel sets of |u|. −1 γ under an additional smallness assumption on M, which controls the norm of f q This would be coherent with known results on the existence of weak solutions. Some results based on rather different duality methods developed in [12] to get Lipschitz regularity, have been obtained in [13]
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