- We are concerned here with well-posed problems for the partial differential
uₜ(x, t) + yMuₜ(x, t) + Lu(x, t) = f(x, t)
containing the elliptic differential operator M of order 2m and the differential
operator L of order ≤2m. Hilbert space methods are used to formulate and solve
an abstract form of the problem and to discuss existence, uniqueness, asymptotic
behavior and boundary conditions of a solution.
The formulation of a generalized problem is the objective of ∮ 1, and we shall
have reason to consider two types of solutions, called weak and strong. Sufficient
conditions on the operator M are given for the existence and uniqueness of a weak
solution to the generalized problem. These conditions constitute elliptic hypotheses
on M and are discussed briefly in ∮ 3. Similar assumptions on L lead to results on
the asymptotic behavior of a weak solution. The case in which M and L are equal
and self-adjoint is discussed in ∮ 2, and it is here that the role of the coefficient y of
the equation appears first. Special as it is, this is a situation that often arises in
applications, and there has been considerable interest in this coefficient y , .
The weak and strong solutions are distinguished not only by regularity conditions
but also by their associated boundary conditions. It first appears in ∮ 5 that it is
possible to prescribe too many (independent) boundary conditions on a strong
solution, but in the applications it is seen that the interdependence of these conditions
is built into the assumptions on the domains of the operators. Two examples of
applications appear in ∮ 6 with a discussion of the types of boundary conditions
that are appropriate.
- Showalter, R. E. (1970). Well-posed problems for a partial differential equation of order 2m+1. SIAM Journal on Mathematical Analysis, 1(2), 214-231. doi:10.1137/0501020
- The published article can be found at the SIAM Journal on Mathematical Analysis (Society for Industrial and Applied Mathematics).
- This is the publisher’s final pdf.
- description.provenance : Made available in DSpace on 2014-11-05T21:53:38Z (GMT). No. of bitstreams: 1ShowalterRalphMathematicsWell-PosedProblems.pdf: 1615060 bytes, checksum: 627ae075aa97387f24dcd34c7080cf90 (MD5) Previous issue date: 1970-05
- description.provenance : Approved for entry into archive by Erin Clark(firstname.lastname@example.org) on 2014-11-05T21:53:38Z (GMT) No. of bitstreams: 1ShowalterRalphMathematicsWell-PosedProblems.pdf: 1615060 bytes, checksum: 627ae075aa97387f24dcd34c7080cf90 (MD5)
- description.provenance : Submitted by Erin Clark (email@example.com) on 2014-11-05T21:53:28ZNo. of bitstreams: 1ShowalterRalphMathematicsWell-PosedProblems.pdf: 1615060 bytes, checksum: 627ae075aa97387f24dcd34c7080cf90 (MD5)