Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

A boundary Harnack inequality for singular equations of $ p$-parabolic type


Authors: Tuomo Kuusi, Giuseppe Mingione and Kaj Nyström
Journal: Proc. Amer. Math. Soc. 142 (2014), 2705-2719
MSC (2010): Primary 35K10, 35K67, 35K92, 35B65
DOI: https://doi.org/10.1090/S0002-9939-2014-12171-X
Published electronically: April 29, 2014
MathSciNet review: 3209326
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove a boundary Harnack type inequality for nonnegative solutions to singular equations of $ p$-parabolic type, $ 2n/(n+1)<p<2$, in a time-independent cylinder whose base is $ C^{1,1}$-regular. Simple examples show, using the corresponding estimates valid for the heat equation as a point of reference, that this type of inequality cannot, in general, be expected to hold in the degenerate case ( $ 2<p<\infty $).


References [Enhancements On Off] (What's this?)

  • [1] Emilio Acerbi and Giuseppe Mingione, Gradient estimates for a class of parabolic systems, Duke Math. J. 136 (2007), no. 2, 285-320. MR 2286632 (2007k:35211), https://doi.org/10.1215/S0012-7094-07-13623-8
  • [2] Hiroaki Aikawa, Tero Kilpeläinen, Nageswari Shanmugalingam, and Xiao Zhong, Boundary Harnack principle for $ p$-harmonic functions in smooth Euclidean domains, Potential Anal. 26 (2007), no. 3, 281-301. MR 2286038 (2008a:31012), https://doi.org/10.1007/s11118-006-9036-y
  • [3] B. Avelin, U. Gianazza and S. Salsa, Boundary estimates for certain degenerate and singular parabolic equations, Preprint (2012).
  • [4] Emmanuele DiBenedetto, Degenerate parabolic equations, Universitext, Springer-Verlag, New York, 1993. MR 1230384 (94h:35130)
  • [5] Emmanuele DiBenedetto and Avner Friedman, Hölder estimates for nonlinear degenerate parabolic systems, J. Reine Angew. Math. 357 (1985), 1-22. MR 783531 (87f:35134a), https://doi.org/10.1515/crll.1985.357.1
  • [6] Emmanuele DiBenedetto, Ugo Gianazza, and Vincenzo Vespri, Harnack estimates for quasi-linear degenerate parabolic differential equations, Acta Math. 200 (2008), no. 2, 181-209. MR 2413134 (2009g:35130), https://doi.org/10.1007/s11511-008-0026-3
  • [7] Emmanuele DiBenedetto, Ugo Gianazza, and Vincenzo Vespri, Forward, backward and elliptic Harnack inequalities for non-negative solutions to certain singular parabolic partial differential equations, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 9 (2010), no. 2, 385-422. MR 2731161 (2011j:35121)
  • [8] Eugene B. Fabes, Nicola Garofalo, and Sandro Salsa, A backward Harnack inequality and Fatou theorem for nonnegative solutions of parabolic equations, Illinois J. Math. 30 (1986), no. 4, 536-565. MR 857210 (88d:35089)
  • [9] E. B. Fabes and M. V. Safonov, Behavior near the boundary of positive solutions of second order parabolic equations, Proceedings of the conference dedicated to Professor Miguel de Guzmán (El Escorial, 1996), Special Issue, 1997, pp. 871-882. MR 1600211 (99d:35071), https://doi.org/10.1007/BF02656492
  • [10] E. B. Fabes, M. V. Safonov, and Yu Yuan, Behavior near the boundary of positive solutions of second order parabolic equations. II, Trans. Amer. Math. Soc. 351 (1999), no. 12, 4947-4961. MR 1665328 (2000c:35085), https://doi.org/10.1090/S0002-9947-99-02487-3
  • [11] Nicola Garofalo, Second order parabolic equations in nonvariational forms: boundary Harnack principle and comparison theorems for nonnegative solutions, Ann. Mat. Pura Appl. (4) 138 (1984), 267-296. MR 779547 (87f:35115), https://doi.org/10.1007/BF01762548
  • [12] David S. Jerison and Carlos E. Kenig, Boundary behavior of harmonic functions in nontangentially accessible domains, Adv. in Math. 46 (1982), no. 1, 80-147. MR 676988 (84d:31005b), https://doi.org/10.1016/0001-8708(82)90055-X
  • [13] T. Kilpeläinen and P. Lindqvist, On the Dirichlet boundary value problem for a degenerate parabolic equation, SIAM J. Math. Anal. 27 (1996), no. 3, 661-683. MR 1382827 (97b:35118), https://doi.org/10.1137/0527036
  • [14] Juha Kinnunen and John L. Lewis, Higher integrability for parabolic systems of $ p$-Laplacian type, Duke Math. J. 102 (2000), no. 2, 253-271. MR 1749438 (2001b:35152), https://doi.org/10.1215/S0012-7094-00-10223-2
  • [15] Tuomo Kuusi and Giuseppe Mingione, Potential estimates and gradient boundedness for nonlinear parabolic systems, Rev. Mat. Iberoam. 28 (2012), no. 2, 535-576. MR 2916967
  • [16] T. Kuusi and G. Mingione, Gradient regularity for nonlinear parabolic equations, Ann. Sc. Norm. Super. Pisa Cl. Sci. (V) 12 (2013), 755-822.
  • [17] Tuomo Kuusi, Giuseppe Mingione, and Kaj Nyström, Sharp regularity for evolutionary obstacle problems, interpolative geometries and removable sets, J. Math. Pures Appl. (9) 101 (2014), no. 2, 119-151. MR 3158698, https://doi.org/10.1016/j.matpur.2013.03.004
  • [18] John L. Lewis and Kaj Nyström, Boundary behaviour for $ p$ harmonic functions in Lipschitz and starlike Lipschitz ring domains, Ann. Sci. École Norm. Sup. (4) 40 (2007), no. 5, 765-813 (English, with English and French summaries). MR 2382861 (2009d:35093), https://doi.org/10.1016/j.ansens.2007.09.001
  • [19] John Lewis and Kaj Nyström, Boundary behavior and the Martin boundary problem for $ p$ harmonic functions in Lipschitz domains, Ann. of Math. (2) 172 (2010), no. 3, 1907-1948. MR 2726103 (2011k:31008), https://doi.org/10.4007/annals.2010.172.1907
  • [20] John L. Lewis and Kaj Nyström, Regularity and free boundary regularity for the $ p$ Laplacian in Lipschitz and $ C^1$ domains, Ann. Acad. Sci. Fenn. Math. 33 (2008), no. 2, 523-548. MR 2431379 (2010b:35500)
  • [21] John L. Lewis and Kaj Nyström, Boundary behaviour of $ p$-harmonic functions in domains beyond Lipschitz domains, Adv. Calc. Var. 1 (2008), no. 2, 133-170. MR 2427450 (2009i:35111), https://doi.org/10.1515/ACV.2008.005
  • [22] John L. Lewis and Kaj Nyström, Regularity of Lipschitz free boundaries in two-phase problems for the $ p$-Laplace operator, Adv. Math. 225 (2010), no. 5, 2565-2597. MR 2680176 (2012c:35486), https://doi.org/10.1016/j.aim.2010.05.005
  • [23] John L. Lewis and Kaj Nyström, Regularity of flat free boundaries in two-phase problems for the $ p$-Laplace operator, Ann. Inst. H. Poincaré Anal. Non Linéaire 29 (2012), no. 1, 83-108. MR 2876248, https://doi.org/10.1016/j.anihpc.2011.09.002
  • [24] John L. Lewis and Kaj Nyström, Regularity and free boundary regularity for the $ p$-Laplace operator in Reifenberg flat and Ahlfors regular domains, J. Amer. Math. Soc. 25 (2012), no. 3, 827-862. MR 2904575, https://doi.org/10.1090/S0894-0347-2011-00726-1
  • [25] Gary M. Lieberman, Boundary and initial regularity for solutions of degenerate parabolic equations, Nonlinear Anal. 20 (1993), no. 5, 551-569. MR 1207530 (94e:35041), https://doi.org/10.1016/0362-546X(93)90038-T
  • [26] Kaj Nyström, The Dirichlet problem for second order parabolic operators, Indiana Univ. Math. J. 46 (1997), no. 1, 183-245. MR 1462802 (98k:35087), https://doi.org/10.1512/iumj.1997.46.1277
  • [27] Sandro Salsa, Some properties of nonnegative solutions of parabolic differential operators, Ann. Mat. Pura Appl. (4) 128 (1981), 193-206 (English, with Italian summary). MR 640782 (83j:35078), https://doi.org/10.1007/BF01789473
  • [28] Mikhail V. Safonov and Yu Yuan, Doubling properties for second order parabolic equations, Ann. of Math. (2) 150 (1999), no. 1, 313-327. MR 1715327 (2000h:35059), https://doi.org/10.2307/121104
  • [29] Giuseppe Savaré and Vincenzo Vespri, The asymptotic profile of solutions of a class of doubly nonlinear equations, Nonlinear Anal. 22 (1994), no. 12, 1553-1565. MR 1285092 (96a:35084), https://doi.org/10.1016/0362-546X(94)90188-0

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 35K10, 35K67, 35K92, 35B65

Retrieve articles in all journals with MSC (2010): 35K10, 35K67, 35K92, 35B65


Additional Information

Tuomo Kuusi
Affiliation: Aalto University, Institute of Mathematics, P.O. Box 11100, FI-00076 Aalto, Finland
Email: tuomo.kuusi@aalto.fi

Giuseppe Mingione
Affiliation: Dipartimento di Matematica e Informatica, Università di Parma, Parco Area delle Scienze 53/a, Campus, 43124 Parma, Italy
Email: giuseppe.mingione@unipr.it

Kaj Nyström
Affiliation: Department of Mathematics, Uppsala University, S-751 06 Uppsala, Sweden
Email: kaj.nystrom@math.uu.se

DOI: https://doi.org/10.1090/S0002-9939-2014-12171-X
Keywords: $p$-parabolic equation, degenerate, singular, Harnack inequality, boundary Harnack inequality, Lipschitz domain $C^{1,\alpha}$-domain, $C^{1, 1}$
Received by editor(s): August 29, 2012
Published electronically: April 29, 2014
Communicated by: Tatiana Toro
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society