Overdetermined Neumann boundary value problem in unbounded domains

V. V. Volchkov   ; Vit. V. Volchkov

Ufa mathematical journal, Tome 12 (2020) no. 2, pp. 10-20

Résumé

The studying of overdetermined boundary value problems for elliptic partial differential equations was initiated by J. Serrin in 1971. In his work, he established a property of radial symmetry for solutions of some overdetermined Poisson problem. Apart of a significant independent interest, the problems of such kind have important applications in the potential theory, integral geometry, hydrodynamics and capillarity theory. Usually, the resolving of these problems is based on Hopf lemma on an angular boundary point and the method of hyperplanes motion introduced by A.A. Alexandrov for studying some geometric problems related with characterizing the spheres. Among other more modern methods not involving the maximum principle for the considered problems we mention the duality method, the method of volume derivative as well as an integral method. In the present paper we consider an overdetermined Neumann problem for the Laplace equation $\Delta f=0$ in planar unbounded domains. We show that under some conditions, see Theorem 1 in Section 1, such problem is solvable only for the exterior of a ball. A specific feature of Theorem 1 is that in this theorem, for the first time, we obtain an exact condition for the growth of $f$ at infinity. Moreover, as Theorem 2 in Section 2 shows, other conditions in Theorem 1 are also necessary. In contrast to the earlier works, the proof of Theorem 1 employs some boundary properties of conformal mappings, Smirnov theorem on functions in a class $H_p$ and Fejer-Riesz theorem on non-negative trigonometrical polynomials.

Détail

Keywords: overdetermined problems, Neumann problem, harmonic functions, boundary behavior.

Bibliographie

[1] J. Serrin, “A symmetry problem in potential theory”, Arch. Rat. Mech. Anal., 43:1 (1971), 304–318 | DOI | MR | Zbl

[2] L. E. Payne, G. A. Philippin, “On two free boundary problems in potential theory”, J. Math. Anal. Appl., 161:2 (1991), 332–342 | DOI | MR | Zbl

[3] C. A. Berenstein, M. Shahshahani, “Harmonic analysis and the Pompeiu problem”, Amer. J. Math., 105:5 (1983), 1217–1229 | DOI | MR | Zbl

[4] V. V. Volchkov, Integral Geometry and Convolution Equations, Kluwer, Dordrecht, 2003 | MR | Zbl

[5] R. L. Fosdick, J. Serrin, “Rectilinear steady flow of simple fluids”, Proc. R. Soc. Lond. A, 332:1590 (1973), 311–333 | DOI | MR | Zbl

[6] G. A. Philippin, “On a free boundary problem in Electrostatics”, Math. Meth. Appl. Sci., 12:5 (1990), 387–392 | DOI | MR | Zbl

[7] O. Mendez, W. Reichel, “Electrostatic characterization of spheres”, Forum Math., 12:2 (2000), 223–245 | MR | Zbl

[8] B. Sirakov, “Symmetry for exterior elliptic problems and two conjectures in potential theory”, Ann. Inst. Henri Poincaré, Anal. non Linéaire, 18:2 (2001), 135–156 | DOI | MR | Zbl

[9] W. Reichel, “Radial symmetry for an electrostatic, a capillarity and some fully nonlinear overdetermined problems on exterior domains”, Z. Anal. Anwendungen., 15:3 (1996), 619–635 | DOI | MR | Zbl

[10] L. Zalcman, “A bibliographic survey of the Pompeiu problem”, Approximation by solutions of partial differential equations, eds. Fuglede B. et. al., Kluwer, Dordrecht, 1992, 185–194 | DOI | MR

[11] V. V. Volchkov, Vit. V. Volchkov, Offbeat Integral Geometry on Symmetric Spaces, Birkhäuser, Basel, 2013 | MR | Zbl

[12] S. A. Williams, “Analyticity of the boundary for Lipschitz domains without the Pompeiu property”, Indiana Univ. Math. J., 30:3 (1981), 357–369 | DOI | MR | Zbl

[13] L. A. Caffarelli, L. Karp, H. Shahgholian, “Regularity of a free boundary with application to the Pompeiu problem”, Ann. of Math., 151:2 (2000), 269–292 | DOI | MR | Zbl

[14] C. A. Berenstein, P. Yang, “An overdetermined Neumann problem in the unit disk”, Adv. in Math., 44:1 (1982), 1–17 | DOI | MR | Zbl

[15] N. B. Willms, G. M. L. Gladwell, “Saddle points and overdetermined problems for the Helmholtz equation”, Z. Angew Math. Phys., 45:1 (1994), 1–26 | DOI | MR | Zbl

[16] P. W. Schaefer, “On nonstandard overdetermined boundary value problems”, Nonlinear Analysis, 47:4 (2001), 2203–2212 | DOI | MR | Zbl

[17] N. Garofalo, E. Sartori, “Symmetry in exterior boundary value problems for quasilinear elliptic equations via blow-up and a priori estimates”, Adv. Diff. Eqs., 4:2 (1999), 137–161 | MR | Zbl

[18] W. Reichel, “Radial symmetry for elliptic boundary-value problems on exterior domains”, Arch. Rational Mech. Anal., 137:6 (1997), 381–394 | DOI | MR | Zbl

[19] G. A. Philippin, “Applications of the maximum principle to a variety of problems involving elliptic differential equations”, Pitman Res. Notes Math. Ser., 175 (1988), 34–48 | MR | Zbl

[20] A. D. Alexandrov, “A characteristic property of the spheres”, Ann. Mat. Pura Appl., 58:1 (1962), 303–315 | DOI | MR

[21] L. E. Payne, P. W. Schaefer, “Duality theorems in some overdetermined problems”, Math. Methods in the Appl. Sciences, 11:6 (1989), 805–819 | DOI | MR | Zbl

[22] M. Choulli, A. Henrot, “Use of the domain derivative to prove symmetry results in partial differential equations”, Math. Nachr., 192:1 (1998), 91–103 | DOI | MR | Zbl

[23] B. Brandolini, C. Nitsch, P. Salani, C. Trombetti, “Serrin type overdetermined problems: an alternative proof”, Arch. Rational Mech. Anal., 190:2 (2008), 267–280 | DOI | MR | Zbl

[24] G. M. Goluzin, Geometric theory of functions of a complex variable, Transl. Math. Monog., 26, Amer. Math. Soc., Providence, RI, 1969 | MR | MR | Zbl

[25] B. Ya. Levin, Distribution of zeros of entire functions, Amer. Math. Soc., Providence, RI, 1980 | MR