×

Remarks on an overdetermined boundary value problem. (English) Zbl 1166.35353

Summary: We modify and extend proofs of Serrin’s symmetry result for overdetermined boundary value problems from the Laplace-operator to a general quasilinear operator and remove a strong ellipticity assumption in [Philippin, in: Maximum principles and eigenvalue problems in partial differential equations, Proc. Conf., Knoxville/Tenn. 1987, Pitman Res. Notes Math. Ser. 175, 34–48 (1988; Zbl 0658.35012)] and a growth assumption in [N. Garofalo and J. L. Lewis, Am. J. Math. 111, No. 1, 9–33 (1989; Zbl 0681.35016)] on the diffusion coefficient \(A\), as well as a starshapedness assumption on \(\Omega\) in [I. Fragalà et al., Math. Z. 254, No. 1, 117–132 (2006; Zbl 1220.35077)].

MSC:

35N10 Overdetermined systems of PDEs with variable coefficients
35J65 Nonlinear boundary value problems for linear elliptic equations
35B35 Stability in context of PDEs
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Alexandrov A.D. (1962). A characteristic property of the spheres. Ann. Mat. Pura Appl. 58: 303–354 · Zbl 0107.15603
[2] Brock F. and Henrot A. (2002). A symmetry result for an overdetermined elliptic problem using continuous rearrangement and domain derivative. Rend. Circ. Mat. Palermo 51: 375–390 · Zbl 1194.35282
[3] Degiovanni M., Musesti A. and Squassina M. (2003). On the regularity of solutions in the Pucci-Serrin identity. Calculus Variations 18: 317–334 · Zbl 1046.35039
[4] Fragalà I., Gazzola F. and Kawohl B. (2006). Overdetermined boundary value problems with possibly degenerate ellipticity: a geometric approach. Math. Zeitschr. 254: 117–132 · Zbl 1220.35077
[5] Garofalo N. and Lewis J.L. (1989). A symmetry result related to some overdetermined boundary value problems. Am. J. Math. 111: 9–33 · Zbl 0681.35016
[6] Kawohl, B.: Symmetrization - or how to prove symmetry of solutions to partial differential equations. In: Jäger, W., Nečas, J., John, O., Najzar, K., Stara, J. (eds.) Partial Differential Equations, Theory and Numerical Solution. Chapman & Hall CRC Research Notes in Math. 406 London, pp. 214–229 (1999)
[7] Lieberman G. (1988). Boundary regularity for solutions of degenerate elliptic equations. Nonlin. Anal. 12: 1203–1219 · Zbl 0675.35042
[8] Payne L.A. and Philippin G.A. (1979). Some maximum principles for nonlinear elliptic equations in divergence form with applications to capillary surfaces and to surfaces of constant mean curvature. Nonlin. Anal. 3: 193–211 · Zbl 0408.35015
[9] Philippin, G.A.: Applications of the maximum principle to a variety of problems involving elliptic differential equations, In: Schaefer, P.W. Maximum principles and eigenvalue problems in partial differential equations (Knoxville, TN, 1987), Longman Sci. Tech., Pitman Res. Notes Math. Ser., Harlow, 175, pp. 34–48 (1988)
[10] Pohožaev S.J. (1965). Eigenfunctions of the equation {\(\Delta\)}u + {\(\lambda\)} f(u) = 0. Sov. Math. Doklady 6: 1408–1411 · Zbl 0141.30202
[11] Prajapat J. (1998). Serrin’s result for domains with a corner or cusp. Duke Math. J. 91: 29–31 · Zbl 0943.35022
[12] Pucci P. and Serrin J. (1986). A general variational identity. Indiana Univ. Math. J. 35: 681–703 · Zbl 0625.35027
[13] Rellich F. (1940). Darstellung der Eigenwerte von {\(\Delta\)}u + {\(\lambda\)} u = 0 durch ein Randintegral. Math. Zeitschr. 46: 635–636 · JFM 66.0460.01
[14] Serrin J. (1971). A symmetry problem in potential theory. Arch. Ration. Mech. Anal. 43: 304–318 · Zbl 0222.31007
[15] Sperb R. (1981). Maximum Principles and Applications. Academic, · Zbl 0454.35001
[16] Thorbergsson G. (2000). A survey on isoparametric hypersurfaces and their generalizations. In: Dillen, F.J.E. and Verstraelen, L.C.A. (eds) Handbook of differential geometry, vol I., pp 963–995. North-Holland, Amsterdam · Zbl 0979.53002
[17] Weinberger H. (1971). Remark on the preceding paper of Serrin. Arch. Ration. Mech. Anal. 43: 319–320 · Zbl 0222.31008
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.