Mathematische Annalen

, Volume 342, Issue 1, pp 91–124

# The mixed problem in L p for some two-dimensional Lipschitz domains

• Loredana Lanzani
• Luca Capogna
• Russell M. Brown
Article

## Abstract

We consider the mixed problem,
$$\left\{ \begin{array}{ll} \Delta u = 0 \quad & {\rm in }\, \Omega\\ \frac{\partial u }{\partial \nu} = f_N \quad & {\rm on }\, {\rm N} \\ u = f_D \quad & {\rm on}\,D \end{array} \right.$$
in a class of Lipschitz graph domains in two dimensions with Lipschitz constant at most 1. We suppose the Dirichlet data, f D , has one derivative in L p (D) of the boundary and the Neumann data, f N , is in L p (N). We find a p 0 > 1 so that for p in an interval (1, p 0), we may find a unique solution to the mixed problem and the gradient of the solution lies in L p .

35J05

## References

1. 1.
Azzam, A., Kreyszig, E.: On solutions of elliptic equations satisfying mixed boundary conditions. SIAM J. Math. Anal. 13(2), 254–262 (1982)
2. 2.
Bell, S.R.: The Cauchy Transform, Potential Theory, and Conformal Mapping. Studies in Advanced Mathematics. CRC Press, Boca Raton (1992)Google Scholar
3. 3.
Bergh, J., Löfström, J.: Interpolation Spaces. Springer, Heidelberg (1976)
4. 4.
Brown, R.M.: The mixed problem for Laplace’s equation in a class of Lipschitz domains. Commun. Partial Diff. Eqns. 19, 1217–1233 (1994)
5. 5.
Brown, R.M.: The Neumann problem on Lipschitz domains in Hardy spaces of order less than one. Pac. J. Math. 171(2), 389–407 (1995)
6. 6.
Chang, D.C., Krantz, S.G., Stein, E.M.: H p theory on a smooth domain in R n and elliptic boundary value problems. J. Funct. Anal. 114(2), 286–347 (1993)
7. 7.
Coifman, R.R., McIntosh, A., Meyer, Y.: L’intégrale de Cauchy définit un opérateur borné sur L 2 pour les courbes lipschitziennes. Ann. Math. 116, 361–387 (1982)
8. 8.
Coifman, R.R., Weiss, G.: Extensions of Hardy spaces and their use in analysis. Bull. Am. Math. Soc. 83, 569–645 (1976)
9. 9.
Dahlberg, B.E.J.: Estimates of harmonic measure. Arch. Rational Mech. Anal. 65(3), 275–288 (1977)
10. 10.
Dahlberg, B.E.J., Kenig, C.E.: Hardy spaces and the Neumann problem in L p for Laplace’s equation in Lipschitz domains. Ann. Math. 125, 437–466 (1987)
11. 11.
Escauriaza, L.: Personal communication, 2000Google Scholar
12. 12.
Fabes, E.B.: Layer potential methods for boundary value problems on Lipschitz domains. In: Potential Theory—Surveys and Problems, vol 1344 of Lecture notes in math., pp. 55–80. Springer, Heidelberg (1988)Google Scholar
13. 13.
Fabes, E.B., Mendez, O., Mitrea, M.: Boundary layers on Sobolev-Besov spaces and Poisson’s equation for the Laplacian in Lipschitz domains. J. Funct. Anal. 159(2), 323–368 (1998)
14. 14.
Giaquinta, M.: Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, vol 105 of Annals of Mathematics Studies (1983). Princeton University Press, PrincetonGoogle Scholar
15. 15.
Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin (1983)
16. 16.
Grisvard, P.: Elliptic Problems in Nonsmooth Domains. Pitman, London (1985)
17. 17.
Jerison, D.S., Kenig, C.E.: Hardy spaces, A , and singular integrals on chord-arc domains. Math. Scand. 50(2), 221–247 (1982)
18. 18.
Jerison, D.S., Kenig, C.E.: The Neumann problem on Lipschitz domains. Bull. Am. Math. Soc. 4, 203–207 (1982)
19. 19.
Kenig, C.E.: Weighted H p spaces on Lipschitz domains. Am. J. Math. 102(1), 129–163 (1980)
20. 20.
Kenig, C.E.: Harmonic analysis techniques for second order elliptic boundary value problems. Published for the Conference Board of the Mathematical Sciences, Washington (1994)
21. 21.
Kenig, C.E., Ni, W.M.: On the elliptic equation Lu  −  k  +  K exp[2u]  =  0. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 12(2), 191–224 (1985)
22. 22.
Lagnese, J.: Decay of solutions of wave equations in a bounded region with boundary dissipation. J. Diff. Equat. 50(2), 163–182 (1983)
23. 23.
Lieberman, G.M.: Mixed boundary value problems for elliptic and parabolic differential equations of second order. J. Math. Anal. Appl. 113(2), 422–440 (1986)
24. 24.
Maz’ya, V.G., Rossman, J.: L p estimates of solutions to mixed boundary value problems for the Stokes system in polyhedral domains. Preprint, 2006Google Scholar
25. 25.
Maz’ya, V.G., Rossman, J.: Mixed boundary value problems for the Navier-Stokes system in polyhedral domains. Preprint, 2006Google Scholar
26. 26.
Maz’ya, V.G., Rossman, J.: Pointwise estimates for Green’s kernel of a mixed boundary value problem to the Stokes system in a polyhedral cone. Preprint, 2006Google Scholar
27. 27.
Mitrea, I., Mitrea, M.: The Poisson problem with mixed boundary conditions in Sobolev and Besov spaces in nonsmooth domains. Trans. Amer. Math. Soc. (to appear)Google Scholar
28. 28.
Protter, M.H., Weinberger, H.F.: Maximum Principles in Differential Equations. Prentice-Hall Inc., Englewood Cliffs (1967)Google Scholar
29. 29.
Savaré, G.: Regularity and perturbation results for mixed second order elliptic problems. Commun. Partial Diff. Eqns. 22, 869–899 (1997)
30. 30.
Shen, Z.: Weighted estimates in L 2 for Laplace’s equation on Lipschitz domains. Trans. Am. Math. Soc. 357, 2843–2870 (2005)
31. 31.
Stein, E.M.: Harmonic Analysis: Real-variable Methods, Orthogonality and Oscillatory Integrals. Princeton, New Jersey (1993)
32. 32.
Strömberg, J.O., Torchinsky, A.: Weighted Hardy Spaces. Springer, Berlin (1989)
33. 33.
Sykes, J.D.: L p regularity of solutions of the mixed boundary value problem for Laplace’s equation on a Lipschitz graph domain. PhD thesis, University of Kentucky (1999)Google Scholar
34. 34.
Sykes, J.D., Brown, R.M.: The mixed boundary problem in L p and Hardy spaces for Laplace’s equation on a Lipschitz domain. In: Harmonic Analysis and Boundary Value Problems (Fayetteville, AR, 2000), vol 277 of Contemp. Math., pp. 1–18. Amer. Math. Soc., Providence (2001)Google Scholar
35. 35.
Verchota, G.C.: Layer potentials and regularity for the Dirichlet problem for Laplace’s equation on Lipschitz domains. J. Funct. Anal. 59, 572–611 (1984)