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Optimal regularity of solutions to no-sign obstacle-type problems for the sub-Laplacian

Valentino Magnani and Andreas Minne

Vol. 15 (2022), No. 6, 1429–1456

We establish the optimal CH1,1 interior regularity of solutions to

ΔHu = fχ{u0},

where ΔH denotes the sub-Laplacian operator in a stratified group. We assume the weakest regularity condition on f, namely the group convolution f Γ is CH1,1, where Γ is the fundamental solution of ΔH. The CH1,1 regularity is understood in the sense of Folland and Stein. In the classical Euclidean setting, the first seeds of the above problem were already present in the 1991 paper of Sakai and are also related to quadrature domains. As a special instance of our results, when u is nonnegative and satisfies the above equation, we recover the CH1,1 regularity of solutions to the obstacle problem in stratified groups, which was previously established by Danielli, Garofalo and Salsa. Our regularity result is sharp: it can be seen as the subelliptic counterpart of the C1,1 regularity result due to Andersson, Lindgren and Shahgholian.

sub-Laplacian, obstacle problem, subelliptic equations, stratified groups
Mathematical Subject Classification
Primary: 35H20, 35R35
Received: 5 July 2019
Revised: 20 January 2021
Accepted: 11 March 2021
Published: 10 November 2022
Valentino Magnani
Dipartimento di Matematica
Università di Pisa
Andreas Minne
Department of Mathematics
KTH Royal Institute of Technology