Symmetry in a free boundary problem for degenerate parabolic equations on unbounded domains

Garofalo, Nicola; Sartori, Elena

doi:10.1090/S0002-9939-01-05993-7

Symmetry in a free boundary problem for degenerate parabolic equations on unbounded domains
HTML articles powered by AMS MathViewer

by Nicola Garofalo and Elena Sartori PDF

Proc. Amer. Math. Soc. 129 (2001), 3603-3610 Request permission

Abstract:

We use the method of Alexandroff-Serrin to establish the spherical symmetry of the ground domain and of the weak solution to a free boundary problem for a class of quasi-linear parabolic equations in an unbounded cylinder $\Omega \times (0,T)$, where $\Omega = (\mathbb {R}^{n} \backslash \overline {\Omega _{1}})$, with $\Omega _{1}\subset \mathbb R^n$ a simply connected bounded domain. The equations considered are of the type $u_{t} - div (a(u,|Du|)Du) = c(u,|Du|)$, with $a$ modeled on $|Du|^{p-2}$. We consider a solution satisfying the boundary conditions: $u(x,t)=f(t)$ for $(x,t)\in \partial \Omega _{1} \times (O,T)$, and $u(x,0)=0$, $u\rightarrow 0$ as $|x|\rightarrow \infty$. We show that the overdetermined co-normal condition $a(u,|Du|)\frac {\partial u}{\partial \nu }=g(t)$ for $(x,t)\in \partial \Omega _{1} \times (O,T)$, with $g(\overline T) > 0$ for at least one value $\overline T \in (0,T)$, forces the spherical symmetry of the ground domain and of the solution.

References

Giovanni Alessandrini and Nicola Garofalo, Symmetry for degenerate parabolic equations, Arch. Rational Mech. Anal. 108 (1989), no. 2, 161–174. MR 1011556, DOI 10.1007/BF01053461
A. D. Alexandrov, A characteristic property of spheres, Ann. Mat. Pura Appl. (4) 58 (1962), 303–315. MR 143162, DOI 10.1007/BF02413056
C. J. Everett Jr., Annihilator ideals and representation iteration for abstract rings, Duke Math. J. 5 (1939), 623–627. MR 13
Ya Zhe Chen and Emmanuele DiBenedetto, Boundary estimates for solutions of nonlinear degenerate parabolic systems, J. Reine Angew. Math. 395 (1989), 102–131. MR 983061
Emmanuele DiBenedetto, Degenerate parabolic equations, Universitext, Springer-Verlag, New York, 1993. MR 1230384, DOI 10.1007/978-1-4612-0895-2
Emmanuele DiBenedetto and Avner Friedman, Regularity of solutions of nonlinear degenerate parabolic systems, J. Reine Angew. Math. 349 (1984), 83–128. MR 743967
Emmanuele DiBenedetto and Avner Friedman, Hölder estimates for nonlinear degenerate parabolic systems, J. Reine Angew. Math. 357 (1985), 1–22. MR 783531, DOI 10.1515/crll.1985.357.1
Emmanuele DiBenedetto and Avner Friedman, Hölder estimates for nonlinear degenerate parabolic systems, J. Reine Angew. Math. 357 (1985), 1–22. MR 783531, DOI 10.1515/crll.1985.357.1
Nicola Garofalo and Elena Sartori, Symmetry in exterior boundary value problems for quasilinear elliptic equations via blow-up and a priori estimates, Adv. Differential Equations 4 (1999), no. 2, 137–161. MR 1674355
B. Gidas, Wei Ming Ni, and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979), no. 3, 209–243. MR 544879
Gary M. Lieberman, Boundary regularity for solutions of degenerate parabolic equations, Nonlinear Anal. 14 (1990), no. 6, 501–524. MR 1044078, DOI 10.1016/0362-546X(90)90038-I
Wolfgang Reichel, Radial symmetry for elliptic boundary-value problems on exterior domains, Arch. Rational Mech. Anal. 137 (1997), no. 4, 381–394. MR 1463801, DOI 10.1007/s002050050034
W. Reichel, Radial symmetry for an electrostatic, a capillarity and some fully nonlinear overdetermined problems on exterior domains, Z. Anal. Anwendungen 15 (1996), no. 3, 619–635. MR 1406079, DOI 10.4171/ZAA/719
James Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal. 43 (1971), 304–318. MR 333220, DOI 10.1007/BF00250468