On the logarithmic epiperimetric inequality for the obstacle problem

Luca Spolaor; Bozhidar Velichkov; Luca Spolaor; Bozhidar Velichkov

doi:10.3934/mine.2021004

Mathematics in Engineering

2021, Volume 3, Issue 1: 1-42. doi: 10.3934/mine.2021004

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On the logarithmic epiperimetric inequality for the obstacle problem

Luca Spolaor ¹,
Bozhidar Velichkov ^{2
,
,}

1.
Department of Mathematics, University of California, San Diego, La Jolla, CA, 92093, USA
2.
Dipartimento di Matematica, University of Pisa, Largo B. Pontecorvo 5, 56127 Pisa, Italy

Received: 10 March 2020 Accepted: 13 July 2020 Published: 31 July 2020

We give three different proofs of the log-epiperimetric inequality at singular points for the obstacle problem. In the first, direct proof, we write the competitor explicitly; the second proof is also constructive, but this time the competitor is given through the solution of an evolution problem on the sphere. We compare the competitors obtained in the different proofs and their relation to other similar results that appeared recently. Finally, in the appendix, we give a general theorem, which can be applied also in other contexts and in which the construction of the competitor is reduced to finding a flow satisfying two differential inequalities.
- epiperimetric inequality,
- obstacle problem,
- free boundary,
- singular points
Citation: Luca Spolaor, Bozhidar Velichkov. On the logarithmic epiperimetric inequality for the obstacle problem[J]. Mathematics in Engineering, 2021, 3(1): 1-42. doi: 10.3934/mine.2021004

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Abstract

We give three different proofs of the log-epiperimetric inequality at singular points for the obstacle problem. In the first, direct proof, we write the competitor explicitly; the second proof is also constructive, but this time the competitor is given through the solution of an evolution problem on the sphere. We compare the competitors obtained in the different proofs and their relation to other similar results that appeared recently. Finally, in the appendix, we give a general theorem, which can be applied also in other contexts and in which the construction of the competitor is reduced to finding a flow satisfying two differential inequalities.

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