Lewy-Stampacchia inequality for noncoercive parabolic obstacle problems

Fernando Farroni; Gioconda Moscariello; Gabriella Zecca; Fernando Farroni; Gioconda Moscariello; Gabriella Zecca

doi:10.3934/mine.2023071

Mathematics in Engineering

2023, Volume 5, Issue 4: 1-23. doi: 10.3934/mine.2023071

Previous Article Next Article

Research article Special Issues

Lewy-Stampacchia inequality for noncoercive parabolic obstacle problems

Dipartimento di Matematica e Applicazioni R. Caccioppoli, Università degli Studi di Napoli Federico II, Complesso Universitario di Monte S. Angelo, via Cinthia I-80126 Napoli, Italy

Received: 26 August 2022 Revised: 22 December 2022 Accepted: 22 December 2022 Published: 04 January 2023

We investigate the obstacle problem for a class of nonlinear and noncoercive parabolic variational inequalities whose model is a Leray–Lions type operator having singularities in the coefficients of the lower order terms. We prove the existence of a solution to the obstacle problem satisfying a Lewy-Stampacchia type inequality.
- noncoercive evolution problems,
- obstacle problems,
- penalization,
- Lewy–Stampacchia inequality,
- Marcinkiewicz spaces
Citation: Fernando Farroni, Gioconda Moscariello, Gabriella Zecca. Lewy-Stampacchia inequality for noncoercive parabolic obstacle problems[J]. Mathematics in Engineering, 2023, 5(4): 1-23. doi: 10.3934/mine.2023071

Related Papers:

Abstract

We investigate the obstacle problem for a class of nonlinear and noncoercive parabolic variational inequalities whose model is a Leray–Lions type operator having singularities in the coefficients of the lower order terms. We prove the existence of a solution to the obstacle problem satisfying a Lewy-Stampacchia type inequality.

References

[1]	A. Alvino, Sulla disuguaglianza di Sobolev in Spazi di Lorentz, Boll. Un. Mat. It. A (5), 14 (1977), 148–156.
[2]	H. W. Alt, S. Luckhaus, Quasi-linear elliptic-parabolic differential equations, Math. Z., 183 (1983), 311–341. http://doi.org/10.1007/BF01176474 doi: 10.1007/BF01176474
[3]	A. Bensoussan, J. Lions, Applications of variational inequalities in stochastic control, Amsterdam-New York: North-Holland Publishing, 1982.
[4]	V. Bögelein, F. Duzaar, G. Mingione, Degenerate problems with irregular obstacles, J. Reine Angew. Math., 650 (2011), 107–160. https://doi.org/10.1515/crelle.2011.006 doi: 10.1515/crelle.2011.006
[5]	H. Brézis, Problémes unilatéraux, J. Math. Pures Appl. (9), 51 (1972), 1–168.
[6]	M. Carozza, C. Sbordone, The distance to $L^\infty$ in some function spaces and applications, Differential Integral Equations, 10 (1997), 599–607. https://doi.org/10.57262/die/1367438633 doi: 10.57262/die/1367438633
[7]	F. Donati, A penalty method approach to strong solutions of some nonlinear parabolic unilateral problems, Nonlinear Anal., 6 (1982), 585–597. https://doi.org/10.1016/0362-546X(82)90050-5 doi: 10.1016/0362-546X(82)90050-5
[8]	W. Fang, K. Ito, Weak solutions for diffusion-convection equations, Appl. Math. Lett., 13 (2000), 69–75. https://doi.org/10.1016/S0893-9659(99)00188-3 doi: 10.1016/S0893-9659(99)00188-3
[9]	F. Farroni, G. Moscariello, A nonlinear parabolic equation with drift term, Nonlinear Anal., 177 (2018), 397–412. https://doi.org/10.1016/j.na.2018.04.021 doi: 10.1016/j.na.2018.04.021
[10]	F. Farroni, L. Greco, G. Moscariello, G. Zecca, Noncoercive quasilinear elliptic operators with singular lower order terms, Calc. Var., 60 (2021), 83. https://doi.org/10.1007/s00526-021-01965-z doi: 10.1007/s00526-021-01965-z
[11]	F. Farroni, L. Greco, G. Moscariello, G. Zecca, Nonlinear evolution problems with singular coefficients in the lower order terms, Nonlinear Differ. Equ. Appl., 28 (2021), 38. https://doi.org/10.1007/s00030-021-00698-4 doi: 10.1007/s00030-021-00698-4
[12]	F. Farroni, L. Greco, G. Moscariello, G. Zecca, Noncoercive parabolic obstacle problems, preprint.
[13]	N. Gigli, S. Mosconi, The abstract Lewy–Stampacchia inequality and applications, J. Math. Pure. Appl., 104 (2015), 258–275. https://doi.org/10.1016/j.matpur.2015.02.007 doi: 10.1016/j.matpur.2015.02.007
[14]	L. Greco, G. Moscariello, G. Zecca, An obstacle problem for noncoercive operators, Abstr. Appl. Anal., 2015 (20105), 890289. https://doi.org/10.1155/2015/890289
[15]	O. Guibé, A. Mokrane, Y. Tahraoui, G. Vallet, Lewy-Stampacchia's inequality for a pseudomonotone parabolic problem, Adv. Nonlinear Anal., 9 (2020), 591–612. https://doi.org/10.1515/anona-2020-0015 doi: 10.1515/anona-2020-0015
[16]	D. Kinderlehrer, G. Stampacchia, An introduction to variational inequalities and their applications, New York: Academic Press, 1980. https://doi.org/10.1137/1.9780898719451
[17]	J. Korvenpää, T. Kuusi, G. Palatucci, The obstacle problem for nonlinear integro-differential operators, Calc. Var., 55 (2016), 63. https://doi.org/10.1007/s00526-016-0999-2 doi: 10.1007/s00526-016-0999-2
[18]	T. Kuusi, G. Mingione, K. Nyström, Sharp regularity for evolutionary obstacle problems, interpolative geometries and removable sets, J. Math. Pure. Appl., 101 (2014), 119–151. https://doi.org/10.1016/j.matpur.2013.03.004 doi: 10.1016/j.matpur.2013.03.004
[19]	H. Lewy, G. Stampacchia, On the regularity of the solution of a variational inequality, Commun. Pure Appl. Math., 22 (1969), 153–188. https://doi.org/10.1002/cpa.3160220203 doi: 10.1002/cpa.3160220203
[20]	J.-L. Lions, G. Stampacchia, Variational inequalities, Commun. Pure Appl. Math., 20 (1967), 493–519. https://doi.org/10.1002/cpa.3160200302
[21]	J. Leray, J. L. Lions, Quelques résultats de Visik sur le problèmes elliptiques non linéaires par les méthodes de Minty-Browder, Bull. Soc. Math. France, 93 (1965), 97–107. https://doi.org/10.24033/bsmf.1617 doi: 10.24033/bsmf.1617
[22]	G. Mingione, G. Palatucci, Developments and perspectives in nonlinear potential theory, Nonlinear Anal., 194 (2020), 111452. https://doi.org/10.1016/j.na.2019.02.006 doi: 10.1016/j.na.2019.02.006
[23]	A. Mokrane, F. Murat, A proof of the Lewy-Stampacchia's inequality by a penalization method, Potential Anal., 9 (1998), 105–142. https://doi.org/10.1023/A:1008649609888 doi: 10.1023/A:1008649609888
[24]	R. O'Neil, Convolutions operators and $L(p, q)$ spaces, Duke Math. J., 30 (1963), 129–142. https://doi.org/10.1215/S0012-7094-63-03015-1 doi: 10.1215/S0012-7094-63-03015-1
[25]	J.-F. Rodrigues, Obstacle problems in mathematical physics, Amsterdam: North-Holland Publishing Co., 1987.
[26]	R. Servadei, E. Valdinoci, Lewy-Stampacchia type estimates for variational inequalities driven by (non)local operators, Rev. Mat. Iberoam., 29 (2013), 1091–1126. https://doi.org/10.4171/RMI/750 doi: 10.4171/RMI/750
[27]	R. E. Showalter, Monotone operators in Banach space and nonlinear partial differential equations, Providence, RI: American Mathematical Society, 1997.

Reader Comments

Your name:*

Email:*
© 2023 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)