Regularity results for a penalized boundary obstacle problem

Donatella Danielli; Rohit Jain; Donatella Danielli; Rohit Jain

doi:10.3934/mine.2021007

Mathematics in Engineering

2021, Volume 3, Issue 1: 1-23. doi: 10.3934/mine.2021007

Previous Article Next Article

Research article Special Issues

Regularity results for a penalized boundary obstacle problem

Donatella Danielli ^{1
,
,},
Rohit Jain ²

1.
Department of Mathematics, Purdue University, 150 N. University St., West Lafayette, IN 47907, USA
2.
Materiall, 500 E Calaveras, Suite 240, Milpitas, CA 95035, USA

Received: 02 March 2020 Accepted: 13 August 2020 Published: 31 August 2020

In this paper we are concerned with a two-penalty boundary obstacle problem of interest in thermics, fluid dynamics and electricity. Specifically, we prove existence, uniqueness and optimal regularity of the solutions, and we establish structural properties of the free boundary. A central role is played by the monotonicity of ad hoc Almgren- and Monneau-type functionals.
- free boundary problems,
- obstacle problems,
- penalized boundary conditions,
- monotonicity formulas,
- semi-permeable membranes
Citation: Donatella Danielli, Rohit Jain. Regularity results for a penalized boundary obstacle problem[J]. Mathematics in Engineering, 2021, 3(1): 1-23. doi: 10.3934/mine.2021007

Related Papers:

Abstract

In this paper we are concerned with a two-penalty boundary obstacle problem of interest in thermics, fluid dynamics and electricity. Specifically, we prove existence, uniqueness and optimal regularity of the solutions, and we establish structural properties of the free boundary. A central role is played by the monotonicity of ad hoc Almgren- and Monneau-type functionals.

References

[1]	Athanasopoulos I, Caffarelli LA (2006) Optimal regularity of lower dimensional obstacle problems. J Math Sci 132: 274-284. doi: 10.1007/s10958-005-0496-1
[2]	Athanasopoulos I, Caffarelli LA (2010) Continuity of the temperature in boundary heat control problems. Adv Math 224: 293-315. doi: 10.1016/j.aim.2009.11.010
[3]	Athanasopoulos I, Caffarelli LA, Salsa S (2008) The structure of the free boundary for lower dimensional obstacle problems. Am J Math 130: 485-498. doi: 10.1353/ajm.2008.0016
[4]	Allen M, Lindgren E, Petrosyan A (2015) The two-phase fractional obstacle problem. SIAM J Math Anal 47: 1879-1905. doi: 10.1137/140974195
[5]	Allen M, Petrosyan A (2012) A two-phase problem with a lower-dimensional free boundary. Interface Free Bound 14: 307-342. doi: 10.4171/IFB/283
[6]	Caffarelli LA, Salsa S, Silvestre L (2008) Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian. Invent Math 171: 425-461. doi: 10.1007/s00222-007-0086-6
[7]	Caffarelli L, Silvestre L (2007) An extension problem related to the fractional Laplacian. Commun Part Diff Eq 32: 1245-1260. doi: 10.1080/03605300600987306
[8]	Cannarsa P, Sinestrari C (2004) Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control, Boston: Birkhäuser Boston, Inc.
[9]	Danielli D, Krummel B (2020) Existence and regularity results for the penalized thin obstacle problem with variable coefficients. arXiv:2005.05524.
[10]	Danielli D, Salsa S (2017) Obstacle Problems Involving the Fractional Laplacian, Berlin: De Gruyter.
[11]	De Silva D, Savin O (2016) Boundary Harnack estimates in slit domains and applications to thin free boundary problems. Rev Mat Iberoam 32: 891-912. doi: 10.4171/RMI/902
[12]	Duvaut G, Lions JL (1976) Inequalities in Mechanics and Physics, Berlin-New York: SpringerVerlag.
[13]	Fall MM, Felli V (2014) Unique continuation property and local asymptotics of solutions to fractional elliptic equations, Commun Part Diff Eq 39: 354-397.
[14]	Garofalo N, Petrosyan A (2009) Some new monotonicity formulas and the singular set in the lower dimensional obstacle problem. Invent Math 177: 415-461. doi: 10.1007/s00222-009-0188-4
[15]	Jost J (2007) Partial Differential Equations, 2 Eds., New York: Springer.
[16]	Koch H, Petrosyan A, Shi W (2015) Higher regularity of the free boundary in the elliptic Signorini problem. Nonlinear Anal 126: 3-44. doi: 10.1016/j.na.2015.01.007
[17]	Lieberman GM (2013) Oblique Derivative Problems for Elliptic Equations, Hackensack: World Scientific Publishing Co. Pte. Ltd.
[18]	Petrosyan A, Shahgholian H, Uraltseva N (2012) Regularity of Free Boundaries in Obstacle-Type Problems, Providence: American Mathematical Society.

Reader Comments

Your name:*

Email:*
© 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)