Scale invariant regularity estimates for second order elliptic equations\n with lower order coefficients in optimal spaces

2020/05/28 by Georgios Sakellaris, Sakellaris, Georgios
Mathematics · #Advanced Harmonic Analysis Research #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2005.14086

openalex publication_date 2020/05/28 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28

Abstract

We show local and global scale invariant regularity estimates for\nsubsolutions and supersolutions to the equation - rm div(A\∇ν+bu)+c\∇ u+du=- rm divf+g, assuming that A is elliptic and bounded.\nIn the setting of Lorentz spaces, under the assumptions b,f\∈ L^n,1,\nd,g\∈ L^\(n)/(2),1 and c\∈ L^n,q for q\≤\∞, we show that,\nwith the surprising exception of the reverse Moser estimate, scale invariant\nestimates with "good" constants (that is, depending only on the norms of the\ncoefficients) do not hold in general. On the other hand, assuming a necessary\nsmallness condition on b,d or c,d, we show a maximum principle and Moser's\nestimate for subsolutions with "good" constants. We also show the reverse Moser\nestimate for nonnegative supersolutions with "good" constants, under no\nsmallness assumptions when q<\∞, leading to the Harnack inequality for\nnonnegative solutions and local continuity of solutions. Finally, we show that,\nin the setting of Lorentz spaces, our assumptions are the sharp ones to\nguarantee these estimates.\n

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