Kaj Nyström ^[1]

A backward in time Harnack inequality for non-negative solutions to fully non-linear parabolic equations

Pages: 1-14
Received: 1 March 2013   
Accepted: 21 June 2013
Mathematics Subject Classification (2010): 35K55.
Keywords: Fully non-linear parabolic equations, Lipschitz domain, Harnack inequality, backward Harnack inequality.
Author address:
[1] : Department of Mathematics, Uppsala University, S-751 06 Uppsala, Sweden

Abstract: We consider fully non-linear parabolic equations of the form \[ Hu =F(D^2u(x,t),Du(x,t),x,t)-\partial_tu = 0 \] in bounded space-time domains \(D\subset\mathbb R^{n+1}\), assuming only \(F(0,0,x,t)=0\) and a uniform parbolicity condition on \(F\). For domains of the form \(\Omega_T=\Omega\times (0,T)\), where \(\Omega\subset\mathbb R^n\) is a bounded Lipschitz and \(T>0\), we establish a scale-invariant backward in time Harnack inequality for non-negative solutions vanishing on the lateral boundary. Our argument rests on the comparison principle, the Harnack inequality and local Hölder continuity estimates.

References

[ACS] I. Athanasopoulos, L. Caffarelli and S. Salsa, Regularity of the free boundary in parabolic phase-transition problems, Acta Math. 176 (1996), 245-282.
[ACS1] I. Athanasopoulos, L. Caffarelli and S. Salsa, Caloric functions in Lipschitz domains and the regularity of solutions to phase transition problems, Ann. of Math. (2) 143 (1996), 413-434.
[AGS] B. Avelin, U. Gianazza and S. Salsa, Boundary estimates for certain degenerate and singular parabolic equations, preprint.
[BG] A. Banerjee and N. Garofalo, Boundary behavior of nonnegative solutions of fully nonlinear parabolic equations, preprint, arXiv:1212.5985v1 [math.AP].
[CKS] M. G. Crandall, M. Kocan and A. Świech, \(L^p\)-theory for fully nonlinear uniformly parabolic equations, Comm. Partial Differential Equations 25 (2000), 1997-2053.
[FGS] 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 (1986), no. 4, 536-565.
[FS] E. B. Fabes and M. V. Safonov, Behavior near the boundary of positive solutions of second order parabolic equations, J. Fourier Anal. Appl. 3 (1997), no. 1 suppl., 871-882.
[FSY] 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 (1999), 4947-4961.
[G] N. Garofalo, Second order parabolic equations in nonvariational forms: boundary Harnack principle and comparison theorems for nonnegative solutions, Ann. Mat. Pura Appl. 138 (1984), 267-296.
[GGIS] Y. Giga, S. Goto, H. Ishii and M.-H. Sato, Comparison principle and convexity preserving properties for singular degenerate parabolic equations on unbounded domains, Indiana Univ. Math. J. 40 (1991), 443-470.
[HLN] S. Hofmann, J. Lewis and K. Nyström, Caloric measure in parabolic flat domains, Duke Math. J. 122 (2004), 281-346.
[JK] D. S. Jerison and C. E. Kenig, Boundary behavior of harmonic functions in nontangentially accessible domains, Adv. in Math. 46 (1982), 80-147.
[KMN] T. Kuusi, G. Mingione and K. Nyström, A boundary Harnack inequality for singular equations of p-parabolic type, Proc. Amer. Math. Soc., to appear.
[N] K. Nyström, The Dirichlet problem for second order parabolic operators, Indiana Univ. Math. J. 46 (1997), 183-245.
[NPS] K. Nyström, H. Persson and O. Sande, Boundary estimates for non-negative solutions to non-linear parabolic equations, preprint (2013).
[S] S. Salsa, Some properties of nonnegative solutions of parabolic differential operators, Ann. Mat. Pura Appl. 128 (1981), 193-206.
[SY] M. V. Safonov and Y. Yuan, Doubling properties for second order parabolic equations, Ann. of Math. (2) 150 (1999), no. 1, 313-327.
[W1] L. Wang, On the regularity theory of fully nonlinear parabolic equations. I, Comm. Pure Appl. Math. 45 (1992), 27-76.
[W2] L. Wang, On the regularity theory of fully nonlinear parabolic equations. II, Comm. Pure Appl. Math. 45 (1992), 141-178.