Duke Mathematical Journal

Sharp Hodge decompositions, Maxwell's equations, and vector Poisson problems on nonsmooth, three-dimensional Riemannian manifolds

Marius Mitrea

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We solve three basic potential theoretic problems: Hodge decompositions for vector fields, Poisson problems for the Hodge Laplacian, and inhomogeneous Maxwell equations in arbitrary Lipschitz subdomains of a smooth, compact, three-dimensional, Riemannian manifold. In each case we derive sharp estimates on Sobolev-Besov scales and establish integral representation formulas for the solution. The proofs rely on tools from harmonic analysis and algebraic topology, such as Calderón-Zygmund theory and de~Rham theory.

Article information

Duke Math. J. Volume 125, Number 3 (2004), 467-547.

First available: 18 November 2004

Permanent link to this document

Digital Object Identifier

Zentralblatt MATH identifier

Mathematical Reviews number (MathSciNet)

Primary: 31C12: Potential theory on Riemannian manifolds [See also 53C20; for Hodge theory, see 58A14] 35Q60: PDEs in connection with optics and electromagnetic theory 58A14: Hodge theory [See also 14C30, 14Fxx, 32J25, 32S35] 58J32: Boundary value problems on manifolds
Secondary: 31B10, 35J25, 42B20, 46E35: Sobolev spaces and other spaces of "smooth" functions, embedding theorems, trace theorems


Mitrea, Marius. Sharp Hodge decompositions, Maxwell's equations, and vector Poisson problems on nonsmooth, three-dimensional Riemannian manifolds. Duke Mathematical Journal 125 (2004), no. 3, 467--547. doi:10.1215/S0012-7094-04-12322-1. http://projecteuclid.org/euclid.dmj/1100793678.

Export citation


  • J. Bergh and J. Löfström, Interpolation Spaces: An Introduction, Grundlehren Math. Wiss. 223, Springer, Berlin, 1976.
  • P. Bidal and G. de Rham, Les formes différentielles harmoniques, Comment. Math. Helv. 19 (1946), 1--49.
  • M. E. Bogovskiĭ, ``Solutions of some problems of vector analysis, associated with the operators $\mathrmdiv$ and $\mathrmgrad$ (in Russian)'' in Theory of Cubature Formulas and the Application of Functional Analysis to Problems of Mathematical Physics (in Russian), ed. S. V. Uspenskiĭ, Trudy Sem. S. L. Soboleva, No. 1 1980, Akad. Nauk SSSR Sibirsk. Otdel., Inst. Mat., Novosibirsk, 1980, 5--40.
  • A. P. Calderón, The multipole expansion of radiation fields, J. Rational Mech. Anal. 3 (1954), 523--537.
  • --. --. --. --., ``Boundary value problems for the Laplace equation in Lipschitzian domains'' in Recent Progress in Fourier Analysis (El Escorial, Spain, 1983), ed. I. Peral and J. L. Rubio de Francia, North-Holland Math. Stud. 111, North-Holland, Amsterdam, 1985, 33--48.
  • A. P. Calderón and A. Zygmund, On singular integrals, Amer. J. Math. 78 (1956), 289--309.
  • M. Cessenat, Mathematical Methods in Electromagnetism: Linear Theory and Applications, Ser. Adv. Math. Appl. Sci. 41, World Scientific, Inc., River Edge, N.J., 1996.
  • R. R. Coifman and G. Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977), 569--645.
  • D. L. Colton and R. Kress, Integral Equation Methods in Scattering Theory, Pure Appl. Math., Wiley, New York, 1983.
  • B. E. J. Dahlberg, Estimates of harmonic measure, Arch. Rational Mech. Anal. 65 (1977), 272--288.
  • --. --. --. --., On the Poisson integral for Lipschitz and $C^1$-domains, Studia Math. 66 (1979), 13--24.
  • --. --. --. --., $L^q$-Estimates for Green potentials in Lipschitz domains, Math. Scand. 44 (1979), 149--170.
  • B. E. J. Dahlberg and C. E. Kenig, Hardy spaces and the Neumann problem in $L^p$ for Laplace's equation in Lipschitz domains, Ann. of Math. (2) 125 (1987), 437--465.
  • --. --. --. --., ``$L^q$-estimates for the three-dimensional system of elastostatics on Lipschitz domains'' in Analysis and Partial Differential Equations, ed. C. Sadosky, Lecture Notes in Pure and Appl. Math. 122, Dekker, New York, 1990, 621--634.
  • R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, Vol. 3, Springer, Berlin, 1990.
  • G. de Rham, Sur la théorie des formes différentielles harmoniques, Ann. Univ. Grenoble. Sect. Sci. Math. Phys. (N.S.) 22 (1946), 135--152.
  • R. A. DeVore and R. C. Sharpley, Besov spaces on domains in $\mathbbR^d$, Trans. Amer. Math. Soc. 335 (1993), 843--864.
  • M. Dindoš and M. Mitrea, Semilinear Poisson problems in Sobolev-Besov spaces on Lipschitz domains, Publ. Mat. 46 (2002), 353--403.
  • E. B. Fabes, M. Jodeit Jr., and N. M. Rivière, Potential techniques for boundary value problems on $C^1$-domains, Acta Math. 141 (1978), 165--186.
  • E. Fabes, O. Mendez, and M. Mitrea, Boundary layers on Sobolev-Besov spaces and Poisson's equation for the Laplacian in Lipschitz domains, J. Funct. Anal. 159 (1998), 323--368.
  • M. Frazier and B. Jawerth, A discrete transform and decompositions of distribution spaces, J. Funct. Anal. 93 (1990), 34--170.
  • R. Godement, Topologie algébrique et théorie des faisceaux, Actualités Sci. Indust. 1252, Publ. Math. Univ. Strasbourg 13, Hermann, Paris, 1958.
  • S. Goldberg, Unbounded Linear Operators: Theory and Applications, Dover, New York, 1985.
  • N. V. Grachev and V. G. Maz'ya, Solvability of a boundary integral equation on a polyhedron, Linköping University, Linköping, Sweden, 1991 research report LiTH-MAT-R-91-50.
  • P. Griffiths and J. Harris, Principles of Algebraic Geometry, Wiley Classics Lib., Wiley, New York, 1994.
  • P. Grisvard, Elliptic Problems in Nonsmooth Domains, Monogr. Stud. Math. 24, Pitman, Boston, 1985.
  • W. V. D. Hodge, The Theory and Applications of Harmonic Integrals, Cambridge Univ., Cambridge, England, 1941.
  • T. Iwaniec, Nonlinear differential forms, Lectures in Jyväskylä, report 80, University of Jyväskylä, Finland, 1998.
  • T. Iwaniec and A. Lutoborski, Integral estimates for null Lagrangians, Arch. Rational Mech. Anal. 125 (1993), 25--79.
  • B. Jawerth and M. Mitrea, Higher-dimensional electromagnetic scattering theory on $C^1$ and Lipschitz domains, Amer. J. Math. 117 (1995), 929--963.
  • D. S. Jerison and C. E. Kenig, The Neumann problem on Lipschitz domains, Bull. Amer. Math. Soc. (N.S.) 4 (1981), 203--207.
  • --. --. --. --., The inhomogeneous Dirichlet problem in Lipschitz domains, J. Funct. Anal. 130 (1995), 161--219.
  • A. Jonsson and H. Wallin, Function spaces on subsets of $\mathbbR^n$, Math. Rep. 2 (1984), no. 1.
  • N. Kalton and M. Mitrea, Stability results on interpolation scales of quasi-Banach spaces and applications, Trans. Amer. Math. Soc. 350 (1998), 3903--3922.
  • T. Kato, Perturbation Theory for Linear Operators, 2nd ed., Grundlehren Math. Wiss. 132, Springer, Berlin, 1976.
  • C. E. Kenig, Harmonic Analysis Techniques for Second Order Elliptic Boundary Value Problems, CBMS Reg. Conf. Ser. Math. 83, Amer. Math. Soc., Providence, 1994.
  • K. Kodaira, Harmonic fields in Riemannian manifolds (generalized potential theory), Ann. of Math. (2) 50 (1949), 587--665.
  • W. S. Massey, Singular Homology Theory, Grad. Texts in Math. 70, Springer, New York, 1980.
  • O. Mendez and M. Mitrea, The Banach envelopes of Besov and Triebel-Lizorkin spaces and applications to partial differential equations, J. Fourier Anal. Appl. 6 (2000), 503--531.
  • D. Mitrea and M. Mitrea, Boundary integral methods for harmonic differential forms in Lipschitz domains, Electron. Res. Announc. Amer. Math. Soc. 2 (1996), 92--97.
  • --. --. --. --., ``General second order, strongly elliptic systems in low dimensional nonsmooth manifolds'' in Harmonic Analysis and Boundary Value Problems (Fayetteville, Ark., 2000), Contemp. Math. 277, Amer. Math. Soc., Providence, 2001, 61--86.
  • --. --. --. --., Finite energy solutions of Maxwell's equations and constructive Hodge decompositions on nonsmooth Riemannian manifolds, J. Funct. Anal. 190 (2002), 339--417.
  • D. Mitrea, M. Mitrea, and J. Pipher, Vector potential theory on nonsmooth domains in $\mathbbR^3$ and applications to electromagnetic scattering, J. Fourier Anal. Appl. 3 (1997), 131--192.
  • D. Mitrea, M. Mitrea, and M. Taylor, Layer potentials, the Hodge Laplacian, and global boundary problems in nonsmooth Riemannian manifolds, Mem. Amer. Math. Soc. 150 (2001), no. 713.
  • M. Mitrea, The method of layer potentials in electromagnetic scattering theory on nonsmooth domains, Duke Math. J. 77 (1995), 111--133.
  • M. Mitrea and M. Taylor, Boundary layer methods for Lipschitz domains in Riemannian manifolds, J. Funct. Anal. 163 (1999), 181--251.
  • --. --. --. --., Potential theory on Lipschitz domains in Riemannian manifolds: Hölder continuous metric tensors, Comm. Partial Differential Equations 25 (2000), 1487--1536.
  • --. --. --. --., Potential theory on Lipschitz domains in Riemannian manifolds: Sobolev-Besov space results and the Poisson problem, J. Funct. Anal. 176 (2000), 1--79.
  • --. --. --. --., Potential theory on Lipschitz domains in Riemannian manifolds: $L^p$ Hardy, and Hölder space results, Comm. Anal. Geom. 9 (2001), 369--421.
  • C. B. Morrey Jr., Multiple Integrals in the Calculus of Variations, Grundlehren Math. Wiss. 130, Springer, New York, 1966.
  • C. B. Morrey Jr. and J. Eells Jr., A variational method in the theory of harmonic integrals, Proc. Nat. Acad. Sci. U.S.A. 41 (1955), 391--395.
  • --. --. --. --., A variational method in the theory of harmonic integrals, I, Ann. of Math. (2) 63 (1956), 91--128.
  • C. Müller, Über die Beugung elektromagnetischer Schwingungen an endlichen homogenen Körpern, Math. Ann. 123 (1951), 345--378.
  • --------, Foundations of the mathematical theory of electromagnetic waves, Grundlehren Math. Wiss. 155, Springer, New York, 1969.
  • J. Peetre, New Thoughts on Besov Spaces, Duke Univ. Math. Ser. 1, Dept. of Math., Duke Univ., Durham, 1976.
  • R. Picard, An elementary proof for a compact imbedding result in generalized electromagnetic theory, Math. Z. 187 (1984), 151--164.
  • T. Runst and W. Sickel, Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations, de Gruyter Ser. Nonlinear Anal. Appl. 3, de Gruyter, Berlin, 1996.
  • G. Schwarz, Hodge Decomposition---A Method for Solving Boundary Value Problems, Lecture Notes in Math. 1607, Springer, Berlin, 1995.
  • M. C. Shaw, Hodge theory on domains with conic singularities, Comm. Partial Differential Equations 8 (1983), 65--88.
  • E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Math. Ser. 30, Princeton Univ. Press, Princeton, 1970.
  • M. E. Taylor, Partial Differential Equations, I--III, Appl. Math. Sci. 115--117, Springer, New York, 1996, 1997.,
  • N. Teleman, The index of signature operators on Lipschitz manifolds, Inst. Hautes Études Sci. Publ. Math. 58 (1983), 39--78.
  • H. Triebel, Theory of Function Spaces, Monogr. Math. 78, Birkhäuser, Berlin, 1983.
  • --. --. --. --., Function spaces on Lipschitz domains and on Lipschitz manifolds: Characteristic functions as pointwise multipliers, Rev. Mat. Complut. 15 (2002), 475--524.
  • G. Verchota, Layer potentials and regularity for the Dirichlet problem for Laplace's equation in Lipschitz domains, J. Funct. Anal. 59 (1984), 572--611.
  • F. W. Warner, Foundations of Differentiable Manifolds and Lie Groups, Grad. Texts in Math. 94, Springer, New York, 1983.