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Laplace Equation—Properties of Solutions—Starting Point of Elliptic Theory

In: Methods for Partial Differential Equations

Author

Listed:
  • Marcelo R. Ebert

    (University of São Paulo, Department of Computing and Mathematics)

  • Michael Reissig

    (TU Bergakademie Freiberg, Institute of Applied Analysis)

Abstract

There exists comprehensive literature on the theory of elliptic partial differential equations. One of the simplest elliptic partial differential equations is the Laplace equation. By means of this equation we explain usual properties of solutions. Here we have in mind maximum-minimum principle or regularity properties of classical solutions. On the other hand we explain properties as hypoellipticity or local solvability, too. Both properties are valid even for larger classes than elliptic equations. Moreover, a boundary integral representation for solutions of the Laplace equation shows the connection to potential theory. Boundary value problems of potential theory of first, second and third kind are introduced and relations to the theory of integral equations are described.

Suggested Citation

  • Marcelo R. Ebert & Michael Reissig, 2018. "Laplace Equation—Properties of Solutions—Starting Point of Elliptic Theory," Springer Books, in: Methods for Partial Differential Equations, chapter 0, pages 79-101, Springer.
    • Handle: RePEc:spr:sprchp:978-3-319-66456-9_8
    DOI: 10.1007/978-3-319-66456-9_8
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