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Characterization of balls in terms of Bessel-potential integral equation. (English) Zbl 1229.31005

Summary: For a bounded \(C^{1}\) domain \(\Omega \subset \mathbb R^N\), we consider the Bessel potential
\[ u(x)=\int_\Omega g_\alpha^{}(x-y)\, dy \] for \(2\leq \alpha <N\). We show that \(u=\text{constant}\) on \(\partial \Omega \) if and only if \(\Omega \) is a ball. The more general Bessel-potential integral equation
\[ u(x)=\int_\Omega g^{}_\alpha(x-y)\, h\big(u(y)\big)\, dy \] is also studied. A similar characterization of balls holds under certain assumptions on \(u\) and \(h\big(u(y)\big)\). To establish our main results, we use an integral form of the celebrated moving plane method of A. D. Alexandroff [Ann. Mat. Pura Appl., IV. Ser. 58, 303–315 (1962; Zbl 0107.15603)], J. Serrin [Arch. Ration. Mech. Anal. 43, 304–318 (1971; Zbl 0222.31007)], and B. Gidas, W.-M. Ni and L. Nirenberg [Commun. Math. Phys. 68, 209–243 (1979; Zbl 0425.35020); Adv. Math., Suppl. Stud. 7A, 369-402 (1981; Zbl 0469.35052)] developed by W. Chen, C. Li and B. Ou [Commun. Pure Appl. Math. 59, No. 3, 330–343 (2006; Zbl 1093.45001); corrigendum 59, No. 7, 1064 (2006; Zbl 1093.45001)].

MSC:

31B10 Integral representations, integral operators, integral equations methods in higher dimensions
35N25 Overdetermined boundary value problems for PDEs and systems of PDEs
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[1] Aftalion, A.; Busca, J., Radial symmetry of overdetermined boundary-value problems in exterior domains, Arch. ration. mech. anal., 143, 2, 195-206, (1998) · Zbl 0911.35008
[2] Alexandroff, A.D., A characteristic property of the sphere, Ann. mat. pura appl., 58, 303-354, (1962)
[3] Bennett, A., Symmetry in an overdetermined four order elliptic boundary value problem, SIAM J. math. anal., 17, 1354-1358, (1986) · Zbl 0612.35039
[4] Brandolini, B.; Nitsch, C.; Salani, P.; Trombetti, C., Serrin-type overdetermined problems: an alternative proof, Arch. ration. mech. anal., 190, 267-280, (2008) · Zbl 1161.35025
[5] Caffarelli, L.; Gidas, B.; Spruck, J., Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. pure appl. math., XLII, 271-297, (1989) · Zbl 0702.35085
[6] Chen, W.; Li, C., Methods on nonlinear elliptic equations, AIMS ser. differ. equ. dyn. syst., vol. 4, (2010), AIMS
[7] Chen, W.; Li, C.; Ou, B., Classification of solutions for an integral equation, Comm. pure appl. math., 59, 330-343, (2006) · Zbl 1093.45001
[8] Chang, S.Y.A.; Yang, P.C., On uniqueness of solutions of nth order differential equations in conformal geometry, Math. res. lett., 4, 91-102, (1997)
[9] Cianchi, A.; Salani, P., Overdetermined anisotropic elliptic problems, Math. ann., 345, 4, 859-881, (2009) · Zbl 1179.35107
[10] Fragala, I.; Gazzola, F., Partially overdetermined elliptic boundary value problems, J. differential equations, 245, 1299-1322, (2008) · Zbl 1156.35049
[11] Fragala, I.; Gazzola, F.; Kawohl, B., Overdetermined problems with possibly degenerate ellipticity, a geometric approach, Math. Z., 254, 117-132, (2006) · Zbl 1220.35077
[12] Farina, A.; Kawohl, B., Remarks on an overdetermined boundary value problem, Calc. var. partial differential equations, 31, 3, 351-357, (2008) · Zbl 1166.35353
[13] Farina, A.; Valdinoci, E., Flattening results for elliptic PDEs in unbounded domains with applications to overdetermined problems, Arch. ration. mech. anal., 195, 1025-1058, (2010) · Zbl 1236.35058
[14] Fraenkel, L.E., Introduction to maximum principles and symmetry in elliptic problems, Cambridge tracts in math., vol. 128, (2000), Cambridge University Press London · Zbl 0947.35002
[15] Grafakos, L., Classical and modern Fourier analysis, (2004), Pearson Education, Inc. Upper Saddle River, NJ, xii+931 pp · Zbl 1148.42001
[16] Gidas, B.; Ni, W.; Nirenberg, L., Symmetry of related properties via the maximum principle, Comm. math. phys., 68, 209-243, (1979) · Zbl 0425.35020
[17] Gidas, B.; Ni, W.; Nireberg, L., Symmetry of positive solutions of nonlinear elliptic equations in \(\mathbb{R}^N\), (), 369-402
[18] Garofalo, N.; Lewis, J.L., A symmetry result related to some overdetermined boundary value problems, Amer. J. math., 111, 9-33, (1989) · Zbl 0681.35016
[19] Kenig, C., Harmonic analysis techniques for second order elliptic boundary value problems, CBMS reg. conf. ser. math., vol. 83, (1994), Amer. Math. Soc. Providence, RI
[20] Li, Y., Remark on some conformally invariant integral equations: the method of moving spheres, J. eur. math. soc. (JEMS), 6, 153-180, (2004) · Zbl 1075.45006
[21] Li, D.; Strohmer, G.; Wang, L., Symmetry of integral equations on bounded domain, Proc. amer. math. soc., 137, 3695-3702, (2009) · Zbl 1188.45001
[22] Lu, G.; Zhu, J., An overdetermined problem in Riesz-potential and fractional Laplacian · Zbl 1236.31004
[23] Martensen, E., Eine integralgleichung für die log. gleichgewichtwverteilung und die krümmung der randkurve eines gebies, ZAMM Z. angew. math. mech., 72, 596-599, (1992)
[24] Ma, L.; Chen, D., Radial symmetry and monotonicity for an integral equation, J. math. anal. appl., 342, 943-949, (2009) · Zbl 1140.45004
[25] Payne, L.; Philippin, G., Some overdetermined boundary value problems for harmonic functions, Z. angew. math. phys., 42, 6, 864-873, (1991) · Zbl 0767.35047
[26] Reichel, W., Radial symmetry for elliptic boundary value problems on exterior domain, Arch. ration. mech. anal., 137, 381-394, (1997) · Zbl 0891.35006
[27] Reichel, W., Characterization of balls by Riesz-potentials, Ann. mat., 188, 235-245, (2009) · Zbl 1180.31008
[28] Serrin, J., A symmetry problem in potential theory, Arch. ration. mech. anal., 43, 304-318, (1971) · Zbl 0222.31007
[29] Sirakov, B., Symmetry for exterior elliptic problems and two conjectures in potential theory, Ann. inst. H. Poincaré anal. non linéaire, 18, 135-156, (2001) · Zbl 0997.35014
[30] Stein, E., Singular integrals and differentiability properties of functions, Princeton ser. appl. math., vol. 32, (1970), Princeton Univ. Press Princeton, NJ · Zbl 0207.13501
[31] Wei, J.; Xu, X., Classification of solutions of higher order conformally invariant equations, Math. ann., 313, 2, 207-228, (1999) · Zbl 0940.35082
[32] Wang, G.; Xia, C., A characterization of the Wulff shape by an overdetermined anisotropic PDE, Arch. ration. mech. anal., (2010)
[33] Weinberger, H., Remark on the preceding paper of Serrin, Arch. ration. mech. anal., 43, 319-320, (1971) · Zbl 0222.31008
[34] Ziemer, W., Weakly differentiable function: Sobolev spaces and function of bounded variation, Geom. topol. monogr., vol. 120, (1989), Springer-Verlag New York
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