In this note, we prove that if a subharmonic function Δu ≥ 0 has pure second derivatives ∂iiu that are signed measures, then their negative part (∂iiu)- belongs to L1 (in particular, it is not singular). We then show that this improvement of regularity cannot be upgraded to Lp for any p > 1. We finally relate this problem to a natural question on the one-sided regularity of solutions to the obstacle problem with rough obstacles.
IMPROVED REGULARITY OF SECOND DERIVATIVES FOR SUBHARMONIC FUNCTIONS
Tione R.
2023-01-01
Abstract
In this note, we prove that if a subharmonic function Δu ≥ 0 has pure second derivatives ∂iiu that are signed measures, then their negative part (∂iiu)- belongs to L1 (in particular, it is not singular). We then show that this improvement of regularity cannot be upgraded to Lp for any p > 1. We finally relate this problem to a natural question on the one-sided regularity of solutions to the obstacle problem with rough obstacles.
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