Annali di Matematica Pura ed Applicata

, Volume 157, Issue 1, pp 285–367 | Cite as

Sharp regularity theory for second order hyperbolic equations of Neumann type

Part I. —L2 nonhomogeneous data
  • I. Lasiecka
  • R. Triggiani


We consider the mixed problem for a general, time independent, second order hyperbolic equation in the unknown u, with datum g ε L2(Σ) in the Neumann B.C., with datum f ε L2(Q) in the right hand side of the equation and, say, initial conditions u0=u1=0. We obtain sharp regularity results for u in Q and ù| in ε, by a pseudo-differential approach on the half-space.


Hyperbolic Equation Regularity Result Mixed Problem Regularity Theory Neumann Type 
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  1. [E.1]
    G. Eskin,Paramatrix and propagation of singularities for the interior mixed hyperbolic problem, J. Analyse Math.,32 (1977), pp. 17–62.Google Scholar
  2. [E.2]
    G. Eskin,Initial-boundary value problems for second order hyperbolic equations with general boundary conditions - I, II, J. Analyse Math.,40 (1981), pp. 43–89, and Communic. in P.D.E.,10 (1985), pp. 1117–1212.Google Scholar
  3. [H.1]
    L.Hormander,The analysis of linear partial differential operators, I, II, III, IV, Springer-Verlag, 1983, 1983, 1985, 1985.Google Scholar
  4. [L.1]
    I.Lasiecka,Sharp regularity results for mixed hyperbolic problems of second order, Lectures Notes in Mathematics, 1223, Springer-Verlag, 1986.Google Scholar
  5. [L.2]
    J. L. Lions,Contrôle des systèmes distribués singuliers, Gauthier-Villars, Paris, 1983.Google Scholar
  6. [L-T.1]
    I.Lasiecka - R.Triggiani,A cosine operator approach to modeling L 2(0,T;L 2(Γ))boundary input hyperbolic equations, Appl. Math. & Optimiz., i (1981), pp. 35–83.Google Scholar
  7. [L-T.2]
    I. Lasiecka -R. Triggiani,Regularity of hyperbolic equations under L2(0,T; L 2(Γ))-boundary terms, Appl. Math. & Optimiz.,10 (1983), pp. 275–286.Google Scholar
  8. [L-T.3]
    I. Lasiecka -R. Triggiani,Trace regularity of the solutions of the wave equation with homogeneous Neumann boundary conditions and compactly supported data, J. Math. Anal. & Appl.,141, no. 1 (1989), pp. 49–71.Google Scholar
  9. [L-T.4]
    I. Lasiecka -R. Triggiani,A lifting theorem for the time regularity of solutions to abstract equations with unbounded operators and applications to hyperbolic equations, Proc. Amer. Math. Soc.,104, no. 3 (1988), pp. 745–755.Google Scholar
  10. [L-T.5]
    I.Lasiecka - R.Triggiani,Regularity theory of hyperbolic equations with non-homogeneous Neumann boundary conditions, Part II:General boundary data, preprint 1989.Google Scholar
  11. [L-T.6]
    I.Lasiecka - R.Triggiani,Differential Riccati Equations with unbounded coefficients: applications to boundary control/boundary observation hyperbolic problems, preprint 1990.Google Scholar
  12. [L-T.7]
    I.Lasiecka - R.Triggiani,Announcement in Atti Acc. Lincei Rend. Fis., (8), LXXXIII (1989), to appear.Google Scholar
  13. [L-L-T.1]
    I. Lasiecka -J. L. Lions -R. Triggiani,Nonhomogeneous boundary value problems for second order hyperbolic operators, J. Math. Pures et Appl.,69 (1986), pp. 149–192.Google Scholar
  14. [L-M.1]
    J. L.Lions - E.Magenes,Nonhomogeneous Boundary Value Problems and Applications, I, II (1972) and III (1973), Springer-Verlag.Google Scholar
  15. [M.1]
    S. Myatake,Mixed problem for hyperbolic equations of second order, J. Math. Kyoto Univ.,130, no. 3 (1973), pp. 435–487.Google Scholar
  16. [S.1]
    R. Sakamoto,Mixed problems for hyperbolic equations, I, II, J. Math. Kyoto Univ.,10, no. 2 (1970), pp. 343–373 and10, no. 3 (1970), pp. 403–417.Google Scholar
  17. [S.2]
    R.Sakamoto,Hyperbolic boundary value problems, Cambridge University Press (1982).Google Scholar
  18. [S.3]
    W. W. Symes,A trace theorem for solutions of the wave equation and the remote determination of acoustic sources, Math. Methods in the Applied Sciences,5 (1983), pp. 35–93.Google Scholar
  19. [T.1]
    M. E. Taylor,Pseudo-differential operators, Princeton University Press, Princeton, New Jersey (1981).Google Scholar
  20. [T.2]
    R.Triggiani,An announcement of sharp regularity theory for second order hyperbolic equations of Neumann type, Springer-Verlag Lecture Notes, inControl and Information Sciences,114, pp. 284–288; Proceedings IFIP Conference on optimal control for systems governed by partial differential equations, held at University of Santiago de Campostela, Spain, July 1987Google Scholar

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© Fondazione Annali di Matematica Pura ed Applicata 1990

Authors and Affiliations

  • I. Lasiecka
    • 1
  • R. Triggiani
    • 1
  1. 1.Department of Applied Mathematics, Thornton HallUniversity of VirginiaCharlottesvilleUSA

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