Local porosity of the free boundary in a minimum problem

Yuwei Hu; Jun Zheng; Yuwei Hu; Jun Zheng

doi:10.3934/era.2023277

Electronic Research Archive

2023, Volume 31, Issue 9: 5457-5465. doi: 10.3934/era.2023277

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Local porosity of the free boundary in a minimum problem

Yuwei Hu ¹,
Jun Zheng ^{1,2
,
,}

1.
School of Mathematics, Southwest Jiaotong University, Chengdu 611756, Sichuan, China
2.
Department of Electrical Engineering, Polytechnique Montréal, Montreal H3T 1J4, QC, Canada

Received: 10 April 2023 Revised: 11 July 2023 Accepted: 20 July 2023 Published: 03 August 2023

Given certain set $ \mathcal{K} $ and functions $ q $ and $ h $, we study geometric properties of the set $ \partial\{x\in\Omega:u(x) > 0\} $ for non-negative minimizers of the functional $ \mathcal{J} (u) = \int_{\Omega }^{} \, \left(\frac{1}{p}| \nabla u| ^p+q(u^+)^\gamma +hu\right)\text{d}x $ over $ \mathcal{K} $, where $ {\Omega \subset} \mathbb{R} ^n(n\geq 2) $ is an open bounded domain, $ p\in(1, +\infty) $ and $ \gamma \in (0, 1] $ are constants, $ u^+ $ is the positive part of $ u $ and $ \partial\{x\in\Omega :u(x) > 0\} $ is the so-called free boundary. Such a minimum problem arises in physics and chemistry for $ \gamma = 1 $ and $ \gamma \in(0, 1) $, respectively. Using the comparison principle of $ p $-Laplacian equations, we establish first the non-degeneracy of non-negative minimizers near the free boundary, then prove the local porosity of the free boundary.
- free boundary,
- minimizer,
- porosity,
- non-degeneracy,
- $ p $-Laplacian
Citation: Yuwei Hu, Jun Zheng. Local porosity of the free boundary in a minimum problem[J]. Electronic Research Archive, 2023, 31(9): 5457-5465. doi: 10.3934/era.2023277

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Abstract

Given certain set $ \mathcal{K} $ and functions $ q $ and $ h $, we study geometric properties of the set $ \partial\{x\in\Omega:u(x) > 0\} $ for non-negative minimizers of the functional $ \mathcal{J} (u) = \int_{\Omega }^{} \, \left(\frac{1}{p}| \nabla u| ^p+q(u^+)^\gamma +hu\right)\text{d}x $ over $ \mathcal{K} $, where $ {\Omega \subset} \mathbb{R} ^n(n\geq 2) $ is an open bounded domain, $ p\in(1, +\infty) $ and $ \gamma \in (0, 1] $ are constants, $ u^+ $ is the positive part of $ u $ and $ \partial\{x\in\Omega :u(x) > 0\} $ is the so-called free boundary. Such a minimum problem arises in physics and chemistry for $ \gamma = 1 $ and $ \gamma \in(0, 1) $, respectively. Using the comparison principle of $ p $-Laplacian equations, we establish first the non-degeneracy of non-negative minimizers near the free boundary, then prove the local porosity of the free boundary.

References

[1]	C. Federer, Geometric Measure Theory, Springer-Verlag, New York, 1969.
[2]	L. A. Caffarelli, The obstacle problem revisited, J. Fourier Anal. Appl., 4 (1998), 383–402. https://doi.org/10.1007/BF02498216 doi: 10.1007/BF02498216
[3]	C. Lederman, A free boundary problem with a volume penalization, Ann. Sc. Norm. Super. Pisa Cl. Sci., 23 (1996), 249–300.
[4]	H. W. Alt, D. Phillips, A free boundary problem for semi-linear elliptic equations, J. Reine Angew. Math., 368 (1986), 63–107. https://doi.org/10.1515/crll.1986.368.63 doi: 10.1515/crll.1986.368.63
[5]	D. Phillips, A minimization problem and the regularity of solutions in the presence of a free boundary, Indiana Univ. Math. J., 32 (1983), 1–17.
[6]	D. Phillips, Hausdorff measure estimates of a free boundary for a minimum problem, Commun. Partial Differ. Equations, 8 (1983), 1409–1454. https://doi.org/10.1080/03605308308820309 doi: 10.1080/03605308308820309
[7]	H. Berestycki, L. A. Caffarelli, L. Nirenberg, Uniform estimates for regularization of free boundary problems, Anal. Partial Differ. Equations, 122 (1990), 567–619.
[8]	H. W. Alt, G. Gilardi, The behavior of the free boundary for the dam problem, Ann. Sc. Norm. Super. Pisa Cl. Sci., 9 (1982), 571–626.
[9]	A. Acker, Heat flow inequalities with applications to heat flow optimization problems, SIAM J. Math. Anal., 8 (1977), 604–618. https://doi.org/10.1137/0508048 doi: 10.1137/0508048
[10]	H. W. Alt, L. A. Caffarelli, Existence and regularity for a minimum problem with free boundary, J. Reine Angew. Math., 1981 (1981), 105–144. https://doi.org/10.1515/crll.1981.325.105 doi: 10.1515/crll.1981.325.105
[11]	J. Zheng, L. S. Tavares, A free boundary problem with subcritical exponents in Orlicz spaces, Ann. Mat. Pura Appl., 201 (2022), 695–731. https://doi.org/10.1007/s10231-021-01134-1 doi: 10.1007/s10231-021-01134-1
[12]	J. D. Rossi, P. Y. Wang, The limit as $p\rightarrow \infty$ in a two-phase free boundary problem for the $p$-Laplacian, Interfaces Free Boundaries, 18 (2016), 115–135. https://doi.org/10.4171/IFB/359 doi: 10.4171/IFB/359
[13]	S. Challal, A. Lyaghfouri, J. F. Rodrigues, On the $A$-obstacle problem and the Hausdorff measure of its free boundary, Ann. Mat. Pura Appl., 191 (2012), 113–165. https://doi.org/10.1007/s10231-010-0177-7 doi: 10.1007/s10231-010-0177-7
[14]	S. Challal, A. Lyaghfouri, Porosity of free boundaries in $A$-obstacle problems, Nonlinear Anal. Theory Methods Appl., 70 (2009), 2772–2778. https://doi.org/10.1016/j.na.2008.04.002 doi: 10.1016/j.na.2008.04.002
[15]	L. Karp, T. Kilpeläinen, A. Petrosyan, H. Shahgholian, On the porosity of free boundaries in degenerate variational inequalities, J. Differ. Equations, 164 (2000), 110–117. https://doi.org/10.1006/jdeq.1999.3754 doi: 10.1006/jdeq.1999.3754
[16]	K. Lee, H. Shahgholian, Hausdorff measure and stability for the $p$-obstacle problem ($2 < p < \infty$), J. Differ. Equations, 195 (2003), 14–24. https://doi.org/10.1016/j.jde.2003.06.002 doi: 10.1016/j.jde.2003.06.002
[17]	J. Zheng, B. H. Feng, P. H. Zhao, A remark on the two-phase obstacle-type problem for the $p$-Laplacian, Adv. Calc. Var., 11 (2018), 325–334. https://doi.org/10.1515/acv-2015-0049 doi: 10.1515/acv-2015-0049
[18]	J. Zheng, Z. H. Zhang, P. H. Zhao, Porosity of free boundaries in the obstacle problem for quasilinear elliptic equations, Proc. Math. Sci., 123 (2013), 373–382. https://doi.org/10.1007/s12044-013-0137-4 doi: 10.1007/s12044-013-0137-4
[19]	J. Zheng, P. H. Zhao, A remark on Hausdorff measure in obstacle problems, J. Anal. Appl., 31 (2012), 427–439. https://doi.org/10.4171/ZAA/1467 doi: 10.4171/ZAA/1467
[20]	R. Leit$\tilde{a}$o, E. V. Teixeira, Regularity and geometric estimates for minima of discontinuous functionals, Rev. Mat. Iberoam., 31 (2015), 69–108. https://doi.org/10.4171/rmi/827 doi: 10.4171/rmi/827
[21]	F. Ferrari, C. Lederman, Regularity of flat free boundaries for a $p(x)$-Laplacian problem with right hand side, Nonlinear Anal., 212 (2021). https://doi.org/10.1016/j.na.2021.112444 doi: 10.1016/j.na.2021.112444
[22]	J. E. M. Braga, P. R. P. Regis, On the full regularity of the free boundary for minima of Alt-Caffarelli functionals in Orlicz spaces, Ann. Fenn. Math., 47 (2022), 961–977. https://doi.org/10.54330/afm.120561 doi: 10.54330/afm.120561
[23]	J. R. Ockendon, W. R. Hodgkings, Moving Boundary Problems in Heat Flow and Diffusion, Clarendon Press, Oxford, 1975.
[24]	C. J. Duijn, I. S. Pop, Crystal dissolution and precipitation in porous media: pore scale analysis, J. Reine Angew. Math., 2004 (2004), 171–211. https://doi.org/10.1515/crll.2004.2004.577.171 doi: 10.1515/crll.2004.2004.577.171
[25]	U. Hornung, W. Jäger, Diffusion, convection, adsorption, and reaction of chemicals in porous media, J. Differ. Equations, 92 (1991), 199–225. https://doi.org/10.1016/0022-0396(91)90047-D doi: 10.1016/0022-0396(91)90047-D
[26]	L. Zajíček, On $\sigma$-porous sets in abstract space, Abstr. Appl. Anal., 2005 (2005), 509–534. https://doi.org/10.1155/AAA.2005.509 doi: 10.1155/AAA.2005.509
[27]	L. Zajíček, Porosity and $\sigma $-porosity, Real Anal. Exch., 13 (1987), 314–350. https://doi.org/10.2307/44151885 doi: 10.2307/44151885
[28]	Z. Tan, F. Fang, Orlicz-Sobolev versus Hölder local minimizer and multiplicity results for quasilinear elliptic equations, J. Math. Anal. Appl., 402 (2013), 348–370. https://doi.org/10.1016/j.jmaa.2013.01.029 doi: 10.1016/j.jmaa.2013.01.029
[29]	L. C. Evans, R. F. Gariepy, Measure Theory and Fine Properties of Functions, CRC press, Boca Raton, 1992.
[30]	P. Koskela, S. Rohde, Hausdorff dimension and mean porosity, Math. Ann., 309 (1997), 593–609. https://doi.org/10.1007/s002080050129 doi: 10.1007/s002080050129

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