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Overdetermined problems with possibly degenerate ellipticity, a geometric approach

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Abstract

Given an open bounded connected subset Ω of ℝn, we consider the overdetermined boundary value problem obtained by adding both zero Dirichlet and constant Neumann boundary data to the elliptic equation −div(A(|∇u|)∇u)=1 in Ω. We prove that, if this problem admits a solution in a suitable weak sense, then Ω is a ball. This is obtained under fairly general assumptions on Ω and A. In particular, A may be degenerate and no growth condition is required. Our method of proof is quite simple. It relies on a maximum principle for a suitable P-function, combined with some geometric arguments involving the mean curvature of ∂Ω.

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References

  1. Alexandrov, A.D.: Uniqueness theorems for surfaces in the large. V. (Russian) Vestnik Leningrad. Univ. 13, 5–8 (1958)

    Google Scholar 

  2. Alexandrov, A.D.: A characteristic property of the spheres. Ann. Mat. Pura Appl. 58, 303–354 (1962)

    Article  MathSciNet  Google Scholar 

  3. Benci, V., D'Avenia, P., Fortunato, D., Pisani, L.: Solitons in several space dimensions: Derrick's problem and infinitely many solutions. Arch. Ration. Mech. Anal. 154, 297–324 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  4. Benci, V., Fortunato, D., Pisani, L.: Soliton like solutions of a Lorentz invariant equation in dimension 3. Rev. Math. Phys. 10, 315–344 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  5. Berestycki, H., Nirenberg, L.: On the method of moving planes and the sliding method. Bol. Soc. Brasil. Mat. (N.S.) 22, 1–37 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  6. Brock, F., Henrot, A.: A symmetry result for an overdetermined elliptic problem using continuous rearrangement and domain derivative. Rend. Circ. Mat. Palermo 51, 375–390 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  7. Brothers, J.E., Ziemer, W.P.: Minimal rearrangements of Sobolev functions. J. Reine Angew. Math. 384, 153–179 (1988)

    MATH  MathSciNet  Google Scholar 

  8. Burago, Y.D., Zalgaller, V.A.: Geometric Inequalities. Springer-Verlag, 1980

  9. Cheeger, J.: A lower bound for the smallest eigenvalue of the Laplacian. In: Problems in Analysis (Papers dedicated to Salomon Bochner), pp. 195–199. Princeton Univ. Press, 1970

  10. Choulli, M., Henrot, A.: Use of the domain derivative to prove symmetry results in partial differential equations. Math. Nachr. 192, 91–103 (1998)

    MATH  MathSciNet  Google Scholar 

  11. Crasta, G.: A symmetry problem in the calculus of variations. J. Europ. Math. Soc. 8, 139–154 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  12. Derrick, G.H.: Comments on nonlinear wave equations as models for elementary particles. J. Math. Phys. 5, 1252–1254 (1964)

    Article  MathSciNet  Google Scholar 

  13. Damascelli, L., Pacella, F.: Monotonicity and symmetry results for p-Laplace equations and applications. Adv. Differential Equations 5, 1179–1200 (2000)

    MATH  MathSciNet  Google Scholar 

  14. Fortunato, D., Orsina, L., Pisani, L.: Born-Infeld type equations for electrostatic fields. J. Math. Phys. 43, 5698–5706 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  15. Garofalo, N., Lewis, J.L.: A symmetry result related to some overdetermined boundary value problems. Amer. J. Math. 111, 9–33 (1989)

    MATH  MathSciNet  Google Scholar 

  16. Garofalo, N., Sartori, E.: Symmetry in exterior boundary value problems for quasilinear elliptic equations via blow-up and a priori estimates. Adv. Diff. Eq. 4, 137–161 (1999)

    MATH  MathSciNet  Google Scholar 

  17. Gidas, B., Ni, W.M., Nirenberg, L.: Symmetry and related properties via the maximum principle. Comm. Math. Phys. 68, 209–243 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  18. Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order. Springer, 1977

  19. Heppes, A.: A lower bound for the incircle of a domain. Kzl. MTA Szmitstech. Automat. Kutat Int. Budapest 38, 81–82 (1988)

    MATH  MathSciNet  Google Scholar 

  20. Kawohl, B.: Rearrangements and Convexity of Level Sets in PDE. Springer Lecture Notes in Math. 1150, (1985)

  21. Kawohl, B.: On a family of torsional creep problems. J. Reine Angew. Math. 410, 1–22 (1990)

    MATH  MathSciNet  Google Scholar 

  22. Kawohl, B.: Symmetry or not? Mathematical Intelligencer 20, 16–22 (1998)

  23. 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, 1999, pp. 214–229

  24. Kawohl, B., Fridman, V.: Isoperimetric estimates for the first eigenvalue of the p-Laplace operator and the Cheeger constant. Comment. Math. Univ. Carol. 44, 659–667 (2003)

    MathSciNet  MATH  Google Scholar 

  25. Kawohl, B., Lachand-Robert, T.: Characterization of Cheeger sets for convex subsets of the plane. Pacific J. Math. to appear

  26. Mendez, O., Reichel, W.: Electrostatic characterization of spheres. Forum Math. 12, 223–245 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  27. Montiel, S., Ros, A.: Compact hypersurfaces: the Alexandrov theorem for higher order mean curvatures. In: Differential Geometry, Pitman Monogr. Surveys Pure Appl. Math. 52, Longman Sci. Tech. 1991, pp. 279–296

  28. Payne, L.E., Philippin, G.A.: Some maximum principles for nonlinear elliptic equations in divergence form with applications to capillary surfaces and to surfaces of constant mean curvature. Nonlinear Anal. 3, 193–211 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  29. Pestov, G., Ionin, V.: On the largest possible circle imbedded in a given closed curve. Doklady Akad. Nauk. SSSR 127, 1170–1172 (1959) (in Russian)

    MATH  MathSciNet  Google Scholar 

  30. Pohožaev, S.J.: Eigenfunctions of the equation Δuf(u)=0. Soviet Math. Doklady 6, 1408–1411 (1965)

    Google Scholar 

  31. Prajapat, J.: Serrin's result for domains with a corner or cusp. Duke Math. J. 91, 29–31 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  32. Reichel, W.: Radial symmetry for elliptic boundary value problems on exterior domains. Arch. Ration. Mech. Anal. 137, 381–394 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  33. Serrin, J.: On the strong maximum principle for quasilinear second order differential inequalities. J. Funct. Anal. 5, 184–193 (1970)

    Article  MATH  MathSciNet  Google Scholar 

  34. Serrin, J.: A symmetry problem in potential theory. Arch. Ration. Mech. Anal. 43, 304–318 (1971)

    Article  MATH  MathSciNet  Google Scholar 

  35. Sperb, R.: Maximum Principles and Applications. Academic Press, 1981

  36. Vogel, A.L.: Symmetry and regularity for general regions having a solution to certain overdetermined boundary value problems. Atti Sem. Mat. Fis. Univ. Modena 40, 443–484 (1992)

    MATH  MathSciNet  Google Scholar 

  37. Weinberger, H.: Remark on the preceding paper of Serrin. Arch. Ration. Mech. Anal. 43, 319–320 (1971)

    Article  MATH  MathSciNet  Google Scholar 

  38. Wente, H.C.: The symmetry of sessile and pendent drops. Pacific J. Math. 88, 387–397 (1980)

    MATH  MathSciNet  Google Scholar 

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Correspondence to Bernd Kawohl.

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Fragalà, I., Gazzola, F. & Kawohl, B. Overdetermined problems with possibly degenerate ellipticity, a geometric approach. Math. Z. 254, 117–132 (2006). https://doi.org/10.1007/s00209-006-0937-7

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