Revista Matemática Iberoamericana

On a Parabolic Symmetry Problem

Abstract

In this paper we prove a symmetry theorem for the Green function associated to the heat equation in a certain class of bounded domains $\Omega\subset\mathbb{R}^{n+1}$. For $T>0$, let $\Omega_T=\Omega\cap[\mathbb{R}^n\times (0,T)]$ and let $G$ be the Green function of $\Omega_T$ with pole at $(0,0)\in\partial_p\Omega_T$. Assume that the adjoint caloric measure in $\Omega_T$ defined with respect to $(0,0)$, $\hat\omega$, is absolutely continuous with respect to a certain surface measure, $\sigma$, on $\partial_p\Omega_T$. Our main result states that if $$\frac {d\hat\omega}{d\sigma}(X,t)=\lambda\frac {|X|}{2t}$$ for all $(X,t)\in \partial_p\Omega_T\setminus\{(X,t): t=0\}$ and for some $\lambda>0$, then $\partial_p\Omega_T\subseteq\{(X,t):W(X,t)=\lambda\}$ where $W(X,t)$ is the heat kernel and $G=W-\lambda$ in $\Omega_T$. This result has previously been proven by Lewis and Vogel under stronger assumptions on $\Omega$.

Article information

Source
Rev. Mat. Iberoamericana, Volume 23, Number 2 (2007), 513-536.

Dates
First available in Project Euclid: 26 September 2007

https://projecteuclid.org/euclid.rmi/1190831220

Mathematical Reviews number (MathSciNet)
MR2371436

Zentralblatt MATH identifier
1242.35130

Subjects
Primary: 35K05: Heat equation

Citation

Lewis, John L.; Nyström, Kaj. On a Parabolic Symmetry Problem. Rev. Mat. Iberoamericana 23 (2007), no. 2, 513--536. https://projecteuclid.org/euclid.rmi/1190831220

References

• Athanasopoulos, I., Caffarelli, L. and Salsa, S.: Caloric functions in Lipschitz domains and the regularity of solutions to phase transition problems. Ann. of Math. (2) 143 (1996), no. 3, 413-434.
• Doob, J. L.: Classical potential theory and its probablilistic counterpart. Grundlehren der Mathematischen Wissenschaften 262. Springer-Verlag, New York, 1984.
• Fabes, E. and Safonov, M.: Behavior near the boundary of positive solutions of second order parabolic equations. In Proceedings of the conference dedicated to Professor Miguel de Guzmán (El Escorial, 1996)''. J. Fourier Anal. Appl. 3 (1997), Special Issue, 871-882.
• Fabes, E., Safonov, M. and Yuan, Y.: Behavior near the boundary of positive solutions of second order parabolic equations, II. Trans. Amer. Math. Soc. 351 (1999), no. 12, 4947-4961.
• Hofmann, S.: Parabolic singular integrals of Calderón type, rough operators, and caloric layer potentials. Duke Math. J. 90 (1997), no. 2, 209-259.
• Hofmann, S. and Lewis, J.: $L^2$ Solvability and representation by caloric layer potentials in time-varying domains. Ann. of Math. (2) 144 (1996), no. 2, 349-420.
• Hofmann, S., Lewis, J. and Nyström, K.: Existence of big pieces of graphs for parabolic problems. Ann. Acad. Sci. Fenn. Math. 28 (2003), no. 2, 355-384.
• Hofmann, S., Lewis, J. and Nyström, K.: Caloric measure in parabolic flat domains. Duke Math. J. 122 (2004), no. 2, 281-346.
• Kaufmann, R. and Wu, J. M.: Parabolic measure on domains of class $\mbox Lip_1/2$. Compositio Math. 65 (1988), no. 2, 201-207.
• Lewis, J.: On symmetry and uniform rectifiability arising from some overdetermined elliptic and parabolic boundary conditions. In The $p$-harmonic equation and recent advances in analysis, 175-187. Contemp. Math. 370. Amer. Math. Soc., Providence, 2005.
• Lewis, J. and Murray, M.: The method of layer potentials for the heat equation in time-varying domains. Mem. Amer. Math. Soc. 114 (1995), no. 545.
• Lewis, J. and Silver, J.: Parabolic measure and the Dirichlet problem for the heat equation in two dimensions. Indiana Univ. Math. J. 37 (1988), no. 4, 801-839.
• Lewis, J. and Vogel, A.: On some almost everywhere symmetry theorems. In Nonlinear diffusion equations and their equilibrium states, 3 (Gregynog, 1989), 347-374. Progr. Nonlinear Differential Equations Appl. 7. Birkhäuser, Boston, 1992.
• Lewis, J. and Vogel, A.: On pseudospheres that are quasispheres. Rev. Mat. Iberoamericana 17 (2001), no. 2, 221-255.
• Lewis, J. and Vogel, A.: A symmetry theorem revisited. Proc. Amer. Math. Soc. 130 (2001), no. 2, 443-451.
• Lewis, J. and Vogel, A.: Symmetry theorems and uniform rectifiability. Boundary Value Problems special issue: Harnack Estimates, Positivity and Local Behavior of the Degenerate and Singular Equations'' vol. 2007. Article ID 30190, 59 pages, 2007. doi:10.1155/2007/30190.
• Lewis, J. and Vogel, A.: Uniqueness in a free boundary problem. Comm. Partial Differential Equations 31 (2006), no. 10-12, 1591-1614.
• Nyström, K.: The Dirichlet problem for second order parabolic operators. Indiana Univ. Math. J. 46 (1996), no. 1, 183-245.
• Petrosyan, A.: On existence and uniqueness in a free boundary problem from combustion. Comm. Partial Differential Equations 27 (2002), no. 3-4, 763-789.
• Serrin, J.: A symmetry problem in potential theory. Arch. Rational Mech. Anal. 43 (1971), 304-318.
• Vogel, A.: Symmetry and regularity for general regions having a solution to certain overdetermined boundary value problems. Atti Sem. Mat. Fis. Univ. Modena 40 (1992), no. 2, 443-484.