Abstract
We consider the mixed boundary value problem, or Zaremba’s problem, for the Laplacian in a bounded Lipschitz domain Ω in R n, n ≥ 2. We decompose the boundary \( \partial \Omega= D\cup N\) with D and N disjoint. The boundary between D and N is assumed to be a Lipschitz surface in \(\partial \Omega\). We find an exponent q 0 > 1 so that for p between 1 and q 0 we may solve the mixed problem for L p. Thus, if we specify Dirichlet data on D in the Sobolev space W 1,p(D) and Neumann data on N in L p (N), the mixed problem with data f D and f N has a unique solution and the non-tangential maximal function of the gradient lies in \(L^p( \partial \Omega)\). We also obtain results for p = 1 when the data comes from Hardy spaces.
Change history
01 February 2020
We provide an example to show that a technical estimate in our earlier work, ���The Mixed Problem for the Laplacian in Lipschitz Domains���, is not correct.
References
Adams, D.R., Hedberg, L.I.: Function Spaces and Potential Theory. Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences), vol. 314. Springer-Verlag, Berlin (1996)
Brown, R., Mitrea, I., Mitrea, M., Wright, M.: Mixed boundary value problems for the Stokes system. Trans. Amer. Math. Soc. 362(3), 1211–1230 (2010)
Brown, R.M.: The mixed problem for Laplace’s equation in a class of Lipschitz domains. Comm. Partial Diff. Eqns. 19, 1217–1233 (1994)
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)
Brown, R.M., Mitrea, I.: The mixed problem for the Lamé system in a class of Lipschitz domains. J. Differential Equations 246(7), 2577–2589 (2009)
Caffarelli, L.A., Peral, I.: On W 1,p estimates for elliptic equations in divergence form. Comm. Pure Appl. Math. 51(1), 1–21 (1998)
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)
Coifman, R.R., Weiss, G.: Extensions of Hardy spaces and their use in analysis. Bull. Amer. Math. Soc. 83, 569–645 (1976)
Dahlberg, B.E.J.: Estimates of harmonic measure. Arch. Rational Mech. Anal. 65(3), 275–288 (1977)
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)
De Giorgi, E.: Sulla differenziabilità e l’analiticità delle estremali degli integrali multipli regolari. Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat. 3, 25–43 (1957)
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)
Fabes, E.B., Stroock, D.: The L p-integrability of Green’s functions and fundamental solutions for elliptic and parabolic equations. Duke Math. J. 51, 997–1016 (1984)
Gehring, F.W.: The L p-integrability of the partial derivatives of a quasiconformal mapping. Acta Math. 130, 265–277 (1973)
Giaquinta, M.: Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems. Annals of Mathematics Studies, vol. 105. Princeton University Press, Princeton, NJ (1983)
Giaquinta, M.: Introduction to Regularity Theory for Nonlinear Elliptic Systems. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel (1993)
Giaquinta, M., Modica, G.: Regularity results for some classes of higher order nonlinear elliptic systems. J. Reine Angew. Math. 311/312, 145–169 (1979)
Gröger, K.: A W 1,p-estimate for solutions to mixed boundary value problems for second order elliptic differential equations. Math. Ann. 283(4), 679–687 (1989)
Grüter, M., Widman, K.O.: The Green function for uniformly elliptic equations. Manuscripta Math. 37(3), 303–342 (1982)
Hardy, G.H., Littlewood, J.E.: Some properties of fractional integrals. II. Math. Z. 34(1), 403–439 (1932)
Jerison, D.S., Kenig, C.E.: The Neumann problem on Lipschitz domains. Bull. Amer. Math. Soc. 4, 203–207 (1982)
Kenig, C.E.: Harmonic Analysis Techniques for Second Order Elliptic Boundary Value Problems. Published for the Conference Board of the Mathematical Sciences, Washington, DC (1994)
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)
Kilty, J., Shen, Z.: The L p regularity problem on Lipschitz domains. Trans. Amer. Math. Soc. 363(3), 1241–1264 (2010)
Lanzani, L., Capogna, L., Brown, R.M.: The mixed problem in L p for some two-dimensional Lipschitz domains. Math. Ann. 342(1), 91–124 (2008)
Littman, W., Stampacchia, G., Weinberger, H.: Regular points for elliptic equations with discontinuous coefficients. Ann. della Sc. N. Sup. Pisa 17, 45–79 (1963)
Maz′ya, V.G., Rossmann, J.: Pointwise estimates for Green’s kernel of a mixed boundary value problem to the Stokes system in a polyhedral cone. Math. Nachr. 278(15), 1766–1810 (2005)
Maz′ya, V.G., Rossmann, J.: Schauder estimates for solutions to a mixed boundary value problem for the Stokes system in polyhedral domains. Math. Method Appl. Sci. 29(9), 965–1017 (2006)
Maz′ya, V.G., Rossmann, J.: L p estimates of solutions to mixed boundary value problems for the Stokes system in polyhedral domains. Math. Nachr. 280(7), 751–793 (2007)
Meyers, N.G.: An L p-estimate for the gradient of solutions of second order elliptic divergence equations. Ann. Scuola Norm. Sup. Pisa (3) 17, 189–206 (1963)
Mitrea, D.: Layer potentials and Hodge decompositions in two dimensional Lipschitz domains. Math. Ann. 322(1), 75–101 (2002)
Mitrea, I., Mitrea, M.: The Poisson problem with mixed boundary conditions in Sobolev and Besov spaces in non-smooth domains. Trans. Amer. Math. Soc. 359(9), 4143–4182 (electronic) (2007)
Moser, J.: On Harnack’s theorem for elliptic differential operators. Comm. Pure Appl. Math. 14, 577–591 (1961)
Nash, J.: Continuity of solutions of elliptic and parabolic equations. Amer. J. Math. 80, 931–954 (1958)
Savaré, G.: Regularity and perturbation results for mixed second order elliptic problems. Comm. Partial Diff. Eqns. 22, 869–899 (1997)
Shen, Z.: The L p boundary value problems on Lipschitz domains. Adv. Math. 216(1), 212–254 (2007)
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)
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). Contemp. Math., vol. 277, pp. 1–18. Amer. Math. Soc. Providence, RI (2001)
Taylor, J.L., Ott, K.A., Brown, R.M.: The mixed problem in Lipschitz domains with general decompositions of the boundary. Trans. Amer. Math. Soc. arXiv:1111.1468 [math.AP] (2011)
Venouziou, M., Verchota, G.C.: The mixed problem for harmonic functions in polyhedra of ℝ3. In: Perspectives in Partial Differential Equations, Harmonic Analysis and Applications. Proc. Sympos. Pure Math., vol. 79, pp. 407–423. Amer. Math. Soc. Providence, RI (2008)
Verchota, G.C.: Layer potentials and boundary value problems for Laplace’s equation on Lipschitz domains. PhD thesis, University of Minnesota (1982)
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)
Wright, M.: Boundary value problems for the Stokes system in arbitrary Lipschitz domains. Thesis (Ph.D.), University of Missouri, Columbia. ProQuest LLC, Ann Arbor, MI (2008)
Author information
Authors and Affiliations
Corresponding author
Additional information
Research supported, in part, by the National Science Foundation.
This work was partially supported by a grant from the Simons Foundation (#195075 to Russell Brown).
Rights and permissions
About this article
Cite this article
Ott, K.A., Brown, R.M. The Mixed Problem for the Laplacian in Lipschitz Domains. Potential Anal 38, 1333–1364 (2013). https://doi.org/10.1007/s11118-012-9317-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11118-012-9317-6