Existence and uniqueness for a semilinear elliptic problem on Lipschitz domains in Riemannian manifolds II

Dindoš, Martin

doi:10.1090/S0002-9947-02-03210-5

Existence and uniqueness for a semilinear elliptic problem on Lipschitz domains in Riemannian manifolds II

Author: Martin Dindos
Journal: Trans. Amer. Math. Soc. 355 (2003), 1365-1399
MSC (2000): Primary 35J65, 35B65; Secondary 46E35, 42B20
DOI: https://doi.org/10.1090/S0002-9947-02-03210-5
Published electronically: December 2, 2002
MathSciNet review: 1946396
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Extending our recent work for the semilinear elliptic equation on Lipschitz domains, we study a general second-order Dirichlet problem in $\Omega$ . We improve our previous results by studying more general nonlinear terms with polynomial (and in some cases exponential) growth in the variable . We also study the case of nonnegative solutions.

1. J. Bergh and J. Löfström, Interpolation spaces. An introduction. Grundlehren der Mathematischen Wissenschaften, no. 223, Springer-Verlag, Berlin-New York, 1976.MR 58:2349
2. Z.-Q. Chen, R. J. Williams and Z. Zhao, On the existence of positive solutions for semilinear elliptic equations with singular lower order coefficients and Dirichlet boundary conditions, Math. Ann. 315 (1999), 735-769.MR 2001a:35061
3. M. Dindos, Existence and uniqueness for a semilinear elliptic problem on Lipschitz domains in Riemannian manifolds, Comm. Partial Differential Equations 27 (2002), 219-281.
4. M. Dindos, Hardy spaces and potential theory on $C^{1}$ domains in Riemannian manifolds, Preprint (2001).
5. M. Dindos and M. Mitrea, Semilinear Poisson Problem in Sobolev-Besov spaces on Lipschitz domains, Publicacions Matemàtiques 46 (2002), 353-403.
6. B. Dahlberg, On the Poisson integral for Lipschitz and $C^{1}$ domains, Studia Math. LXVI. (1979), 13-26.MR 81g:31007
7. B. Dahlberg and C. Kenig, Hardy spaces and the Neumann problem in $L^{p}$ for Laplace's equation in Lipschitz domains, Ann. Math. 125 (1987), 437-465.MR 88d:35044
8. B. Dahlberg, C. Kenig and G. Verchota, Boundary value problems for the system of elastostatics on Lipschitz domains, Duke Math. J. 57 (1988), 795-818.MR 90d:35259
9. E. Fabes, M. Jodeit, Jr. and N. Rivère, Potential techniques for boundary value problems on $C^{1}$ -domains, Acta Math. 141 (1978), 165-186.MR 80b:31006
10. E. Fabes, C. Kenig and G. Verchota, The Dirichlet problem for the Stokes system on Lipschitz domains, Duke Math. J. 57 (1988), 769-793.MR 90d:35258
11. D. Gilbarg and N. Trudinger, Elliptic partial differential equations of second order (3rd ed.), Springer-Verlag, Berlin-New York, 1998.MR 2001k:35004
12. V. Isakov and A. I. Nachman, Global uniqueness for a two dimensional semilinear elliptic inverse problem, Trans. Amer. Math. Soc. 347 (1995), 3375-3390.MR 95m:35202
13. Z. Jin, Solvability of Dirichlet problem for semilinear elliptic equations on certain domains, Pacific J. Math. 176 (1996), 117-128.MR 97m:35082
14. F. John and L. Nirenberg, On functions of bounded mean oscillation, Comm. Pure Appl. Math. 14 (1961), 415-426.MR 24:A1348
15. C. Kenig, Harmonic analysis techniques for second order elliptic boundary value problems, CBMS Regional Conference Series in Mathematics, vol. 83, American Mathematical Society, Providence, RI, 1994.MR 96a:35040
16. C. Kenig, Elliptic boundary value problems on Lipschitz domains, Beijing Lectures in Harmonic Analysis (E. Stein, ed.), Princeton University Press, Princeton, 1986, pp. 131-183. MR 88a:35066
17. R. Mazzeo and M. Taylor, Curvature and Uniformization, Israel J. Math. 130 (2002), 323-346.
18. D. Mitrea, M. Mitrea and J. Pipher, Vector potential theory on nonsmooth domains in $\mathbb{R}^{3}$ and applications to electromagnetic scattering, J. Fourier Anal. and Appl. 3 (1997), 131-192.MR 99e:31009
19. D. Mitrea, M. Mitrea and M. Taylor, Layer potentials, the Hodge Laplacian, and global boundary problems in nonsmooth Riemannian manifolds, Memoirs Amer. Math. Soc. 150 (2001), no. 713.MR 2002g:58026
20. M. Mitrea and M. Taylor, Boundary layer methods for Lipschitz domains in Riemannian manifolds, J. Funct. Anal. 163 (1999), 181-251.MR 2000b:35050
21. M. Mitrea and M. Taylor, Potential theory on Lipschitz domains in Riemannian manifolds: $L^{p}$ , Hardy and Hölder space results, Comm. Anal. Geom. 9 (2001), no. 2, 369-421. MR 2002f:31012
22. M. Mitrea and M. Taylor, Potential theory on Lipschitz domains in Riemannian manifolds: Hölder continuous metric tensors, Comm. Partial Differential Equations 25 (2000), 1487-1536. MR 2001h:35040
23. M. Taylor, Partial differential equations, vols. 1-3, Springer-Verlag, New York, 1996.MR 98b:35002b; MR 98b:35003; MR 98k:35001
24. G. Verchota, Layer potentials and regularity for the Dirichlet problem for Laplace's equation in Lipschitz domains, J. Funct. Anal. 59 (1984), 472-611.MR 86e:35038