## Analysis & PDE

### Boundary behavior of solutions to the parabolic $p$-Laplace equation

#### Abstract

We establish boundary estimates for nonnegative solutions to the $p$-parabolic equation in the degenerate range $p>2$. Our main results include new parabolic intrinsic Harnack chains in cylindrical NTA domains together with sharp boundary decay estimates. If the underlying domain is $C1,1$-regular, we establish a relatively complete theory of the boundary behavior, including boundary Harnack principles and Hölder continuity of the ratios of two solutions, as well as fine properties of associated boundary measures. There is an intrinsic waiting-time phenomenon present which plays a fundamental role throughout the paper. In particular, conditions on these waiting times rule out well-known examples of explicit solutions violating the boundary Harnack principle.

#### Article information

Source
Anal. PDE, Volume 12, Number 1 (2019), 1-42.

Dates
Accepted: 10 April 2018
First available in Project Euclid: 16 August 2018

https://projecteuclid.org/euclid.apde/1534384913

Digital Object Identifier
doi:10.2140/apde.2019.12.1

Mathematical Reviews number (MathSciNet)
MR3842908

Zentralblatt MATH identifier
06930183

#### Citation

Avelin, Benny; Kuusi, Tuomo; Nyström, Kaj. Boundary behavior of solutions to the parabolic $p$-Laplace equation. Anal. PDE 12 (2019), no. 1, 1--42. doi:10.2140/apde.2019.12.1. https://projecteuclid.org/euclid.apde/1534384913

#### References

• H. Aikawa, T. Kilpeläinen, N. Shanmugalingam, and X. Zhong, “Boundary Harnack principle for $p$-harmonic functions in smooth Euclidean domains”, Potential Anal. 26:3 (2007), 281–301.
• B. Avelin, “On time dependent domains for the degenerate $p$-parabolic equation: Carleson estimate and Hölder continuity”, Math. Ann. 364:1-2 (2016), 667–686.
• B. Avelin and K. Nyström, “Wolff-potential estimates and doubling of subelliptic $p$-harmonic measures”, Nonlinear Anal. 85 (2013), 145–159.
• B. Avelin, N. L. P. Lundström, and K. Nyström, “Optimal doubling, Reifenberg flatness and operators of $p$-Laplace type”, Nonlinear Anal. 74:17 (2011), 5943–5955.
• B. Avelin, U. Gianazza, and S. Salsa, “Boundary estimates for certain degenerate and singular parabolic equations”, J. Eur. Math. Soc. $($JEMS$)$ 18:2 (2016), 381–424.
• M. F. Bidaut-Véron, “Self-similar solutions of the $p$-Laplace heat equation: the case when $p>2$”, Proc. Roy. Soc. Edinburgh Sect. A 139:1 (2009), 1–43.
• A. Björn, J. Björn, U. Gianazza, and M. Parviainen, “Boundary regularity for degenerate and singular parabolic equations”, Calc. Var. Partial Differential Equations 52:3-4 (2015), 797–827.
• M. Bonforte and J. L. Vázquez, “A priori estimates for fractional nonlinear degenerate diffusion equations on bounded domains”, Arch. Ration. Mech. Anal. 218:1 (2015), 317–362.
• E. DiBenedetto, Degenerate parabolic equations, Springer, 1993.
• E. DiBenedetto, Y. Kwong, and V. Vespri, “Local space-analyticity of solutions of certain singular parabolic equations”, Indiana Univ. Math. J. 40:2 (1991), 741–765.
• E. DiBenedetto, U. Gianazza, and V. Vespri, “Harnack estimates for quasi-linear degenerate parabolic differential equations”, Acta Math. 200:2 (2008), 181–209.
• E. DiBenedetto, U. Gianazza, and V. Vespri, Harnack's inequality for degenerate and singular parabolic equations, Springer, 2012.
• E. B. Fabes and M. V. Safonov, “Behavior near the boundary of positive solutions of second order parabolic equations”, J. Fourier Anal. Appl. 3:Special Issue (1997), 871–882.
• E. B. Fabes, N. Garofalo, and S. Salsa, “Comparison theorems for temperatures in noncylindrical domains”, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. $(8)$ 77:1-2 (1984), 1–12.
• E. B. Fabes, N. Garofalo, and S. Salsa, “A backward Harnack inequality and Fatou theorem for nonnegative solutions of parabolic equations”, Illinois J. Math. 30:4 (1986), 536–565.
• E. B. Fabes, M. V. Safonov, and Y. Yuan, “Behavior near the boundary of positive solutions of second order parabolic equations, II”, Trans. Amer. Math. Soc. 351:12 (1999), 4947–4961.
• N. 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.
• D. S. Jerison and C. E. Kenig, “Boundary behavior of harmonic functions in nontangentially accessible domains”, Adv. in Math. 46:1 (1982), 80–147.
• J. T. Kemper, “Temperatures in several variables: kernel functions, representations, and parabolic boundary values”, Trans. Amer. Math. Soc. 167 (1972), 243–262.
• T. Kilpeläinen and P. Lindqvist, “On the Dirichlet boundary value problem for a degenerate parabolic equation”, SIAM J. Math. Anal. 27:3 (1996), 661–683.
• T. Kilpeläinen and X. Zhong, “Growth of entire $\mathscr{A}$-subharmonic functions”, Ann. Acad. Sci. Fenn. Math. 28:1 (2003), 181–192.
• T. Kuusi, “Harnack estimates for weak supersolutions to nonlinear degenerate parabolic equations”, Ann. Sc. Norm. Super. Pisa Cl. Sci. $(5)$ 7:4 (2008), 673–716.
• T. Kuusi and M. Parviainen, “Existence for a degenerate Cauchy problem”, Manuscripta Math. 128:2 (2009), 213–249.
• T. Kuusi, G. Mingione, and K. Nyström, “A boundary Harnack inequality for singular equations of $p$-parabolic type”, Proc. Amer. Math. Soc. 142:8 (2014), 2705–2719.
• T. Kuusi, P. Lindqvist, and M. Parviainen, “Shadows of infinities”, Ann. Mat. Pura Appl. $(4)$ 195:4 (2016), 1185–1206.
• J. L. Lewis and K. Nyström, “Boundary behaviour for $p$ harmonic functions in Lipschitz and starlike Lipschitz ring domains”, Ann. Sci. École Norm. Sup. $(4)$ 40:5 (2007), 765–813.
• J. Lewis and K. Nyström, “Boundary behavior and the Martin boundary problem for $p$ harmonic functions in Lipschitz domains”, Ann. of Math. $(2)$ 172:3 (2010), 1907–1948.
• G. M. Lieberman, “Boundary and initial regularity for solutions of degenerate parabolic equations”, Nonlinear Anal. 20:5 (1993), 551–569.
• K. Nyström, “The Dirichlet problem for second order parabolic operators”, Indiana Univ. Math. J. 46:1 (1997), 183–245.
• K. Nyström, H. Persson, and O. Sande, “Boundary estimates for non-negative solutions to non-linear parabolic equations”, Calc. Var. Partial Differential Equations 54:1 (2015), 847–879.
• S. Salsa, “Some properties of nonnegative solutions of parabolic differential operators”, Ann. Mat. Pura Appl. $(4)$ 128 (1981), 193–206.
• D. Stan and J. L. Vázquez, “Asymptotic behaviour of the doubly nonlinear diffusion equation $u_t=\Delta_pu^m$ on bounded domains”, Nonlinear Anal. 77 (2013), 1–32.