Annali di Matematica Pura ed Applicata (1923 -)

, Volume 193, Issue 6, pp 1807–1821 | Cite as

Hodge decomposition and solution formulas for some first-order time-dependent parabolic operators with non-constant coefficients

  • R. S. Kraußhar
  • M. M. Rodrigues
  • N. VieiraEmail author


In this paper, we present a Hodge decomposition for the \(L_p\)-space of some parabolic first-order partial differential operators with non-constant coefficients. This is done over different types of domains in Euclidean space \(\mathbb{R }^n\) and on some conformally flat cylinders and the \(n\)-torus associated with different spinor bundles. Initially, we apply a regularization procedure in order to control the non-removable singularities over the hyperplane \(t=0\). Using the setting of Clifford algebras combined with a Witt basis, we introduce some specific integral and projection operators. We present an \(L_p\)-decomposition where one of the components is the kernel of the regularized parabolic Dirac operator with non-constant coefficients. After that, we study the behavior of the solutions and the validity of our results when the regularization parameter tends to zero. To round off, we give some analytic solution formulas for the special context of domains on cylinders and \(n\)-tori.


Schrödinger equation on manifolds Regularized parabolic Dirac operator Hypoelliptic equations Regularization procedure Hodge decomposition 

Mathematics Subject Classification (2000)

30G35 35J10 35C15 



M. M. Rodrigues and N. Vieira were supported by FEDER founds through COMPETE–Operational Program Factors of Competitivy (“Programa Operacional Factores de Competitividade”) and by Portuguese funds through the Center for Research and Development in Mathematics and Applications (University of Aveiro) and the Portuguese Foundation for Science and Technology (“FCT–Fundação para a Ciência e a Tecnologia”), within Project PEst-C/MAT/UI4106/2011 with COMPETE Number FCOMP-01-0124-FEDER-022690. The authors would like to express their gratitude to the referees. Their suggestions and corrections lead to an important improvement in the quality of the paper.


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© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Lehrgebiet Mathematik und ihre Didaktik, Erziehungswissenschaftliche FakultätUniversität ErfurtErfurtGermany
  2. 2.Department of Mathematics, CIDMA, Center for Research and Development in Mathematics and ApplicationsUniversity of AveiroAveiroPortugal

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