Hausdorff measure of the free boundary for the $ p $-obstacle problem with subcritical exponents

Jing Yu; Jun Zheng; Jing Yu; Jun Zheng

doi:10.3934/era.2026393

Electronic Research Archive

2026, Volume 34, Issue 12: 9352-9385. doi: 10.3934/era.2026393

Previous Article Next Article

Research article

Hausdorff measure of the free boundary for the $ p $-obstacle problem with subcritical exponents

Jing Yu ,
Jun Zheng ^,

School of Mathematics, Southwest Jiaotong University, Chengdu 611756, China

Received: 14 February 2026 Revised: 26 May 2026 Accepted: 08 June 2026 Published: 10 November 2026

This paper investigates a class of $ p $-obstacle problems with subcritical exponents having the form
$ \begin{align} \mathrm{div}\left( a(x)|\nabla u|^{p-2}\nabla u\right) = m_1\chi_{\{u>0\}}-m_2u^{\lambda-1}\chi_{\{u>0\}} \ \text{in}\ \Omega, \end{align} $
where $ \Omega $ is a smooth bounded domain in $ \mathbb{R}^N (N \geq 2) $, $ m_1 $ and $ m_2 $ are positive constants, the coefficient function $ a \in C^2(\Omega) $ has a positive lower bound, and $ 2 \leq p < \lambda < p^*: = \frac{Np}{N-p} $ when $ p < N $ and $ N \geq 3 $ or $ 2\leq p < \lambda < +\infty $ when $ N = 2 $. By using the mountain-pass lemma, combined with the penalty method, we first establish the existence of non-negative weak solutions. Then, using the De Giorgi-Nash iteration, we prove the $ L^\infty $ bound and local $ C^{1, \alpha} $ continuity for the solutions. In addition, we prove local porosity of the free boundary based on the optimal growth and non-degeneracy of solutions near the free boundary. Furthermore, by means of Lebesgue measure estimates for gradient level sets, we show that at least one solution corresponds to the free boundary having a locally finite $ (N-1) $-dimensional Hausdorff measure.
- obstacle problem,
- free boundary,
- Hausdorff measure,
- porosity,
- subcritical exponents
Citation: Jing Yu, Jun Zheng. Hausdorff measure of the free boundary for the $ p $-obstacle problem with subcritical exponents[J]. Electronic Research Archive, 2026, 34(12): 9352-9385. doi: 10.3934/era.2026393

Related Papers:

Abstract

This paper investigates a class of $ p $-obstacle problems with subcritical exponents having the form

$ \begin{align} \mathrm{div}\left( a(x)|\nabla u|^{p-2}\nabla u\right) = m_1\chi_{\{u>0\}}-m_2u^{\lambda-1}\chi_{\{u>0\}} \ \text{in}\ \Omega, \end{align} $

where $ \Omega $ is a smooth bounded domain in $ \mathbb{R}^N (N \geq 2) $, $ m_1 $ and $ m_2 $ are positive constants, the coefficient function $ a \in C^2(\Omega) $ has a positive lower bound, and $ 2 \leq p < \lambda < p^*: = \frac{Np}{N-p} $ when $ p < N $ and $ N \geq 3 $ or $ 2\leq p < \lambda < +\infty $ when $ N = 2 $. By using the mountain-pass lemma, combined with the penalty method, we first establish the existence of non-negative weak solutions. Then, using the De Giorgi-Nash iteration, we prove the $ L^\infty $ bound and local $ C^{1, \alpha} $ continuity for the solutions. In addition, we prove local porosity of the free boundary based on the optimal growth and non-degeneracy of solutions near the free boundary. Furthermore, by means of Lebesgue measure estimates for gradient level sets, we show that at least one solution corresponds to the free boundary having a locally finite $ (N-1) $-dimensional Hausdorff measure.

References

[1]	C. Wang, On a class of superlinear obstacle problems, J. Differ. Equations, 434 (2025), 113318. https://doi.org/10.1016/j.jde.2025.113318 doi: 10.1016/j.jde.2025.113318
[2]	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
[3]	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
[4]	P. Zhao, J. Zheng, Remarks on Hausdorff measure and stability for the $p$-obstacle problem $(1 < p < 2)$, Proc. Math. Sci., 122 (2012), 129–137. https://doi.org/10.1007/s12044-012-0053-z doi: 10.1007/s12044-012-0053-z
[5]	J. Zheng, Z. Zhang, P. 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
[6]	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
[7]	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
[8]	Y. W. Hu, J. Zheng, L. S. Tavares, A free boundary problem associated with scalar Ginzburg-Landau equation, Math. Nachr., 299 (2026), 1197–129. https://doi.org/10.1002/mana.70134 doi: 10.1002/mana.70134
[9]	J. Andersson, E. Lindgren, H. Shahgholian, Optimal regularity for the obstacle problem for the $p$-Laplacian, J. Differ. Equations, 259 (2015), 2167–2179. https://doi.org/10.1016/j.jde.2015.03.019 doi: 10.1016/j.jde.2015.03.019
[10]	E. DiBenedetto, Degenerate Parabolic Equations, Springer-Verlag, New York, 1993. https://doi.org/10.1007/978-1-4612-0895-2
[11]	H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer Science & Business Media, New York, 2011. https://doi.org/10.1007/978-0-387-70914-7
[12]	O. A. Ladyzhenskaya, N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York, 1968.
[13]	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
[14]	J. Zheng, B. H. Feng, P. H. Zhao, Regularity of minimizers in the two-phase free boundary problems in Orlicz-Sobolev spaces, Z. Anal. Anwend., 36 (2017), 37–47. https://doi.org/10.4171/ZAA/1578 doi: 10.4171/ZAA/1578
[15]	R. Leit$\tilde{a}$o, O. S. de Queiroz, E. V. Teixeira, Regularity for degenerate two-phase free boundary problems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), 741–762. https://doi.org/10.1016/j.anihpc.2014.03.004 doi: 10.1016/j.anihpc.2014.03.004
[16]	A. Ambrosetti, P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349–381. https://doi.org/10.1016/0022-1236(73)90051-7 doi: 10.1016/0022-1236(73)90051-7
[17]	G. M. Lieberman, The natural generalization of the natural conditions of Ladyzhenskaya and Ural'tseva for elliptic equations, Commun. Partial Differ. Equations, 16 (1991), 311–361. https://doi.org/10.1080/03605309108820761 doi: 10.1080/03605309108820761
[18]	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
[19]	O. Martio, M. Vuorinen, Whitney cubes, $p$-capacity, and Minkowski content, Expositiones Math., 5 (1987), 17–40.
[20]	Y. W. Hu, J. Zheng, Porosity of the free boundary in a minimum problem, J. Nonlinear Var. Anal., 8 (2024), 109–124. https://doi.org/10.23952/jnva.8.2024.1.06 doi: 10.23952/jnva.8.2024.1.06
[21]	L. C. Evans, Partial Differential Equations, American Mathematical Society, 2022.
[22]	L. C. Evans, Measure Theory and Fine Properties of Functions, Routledge, New York, 2018. https://doi.org/10.1201/9780203747940

Reader Comments

Your name:*

Email:*
© 2026 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)