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Boundary Value Problems for Partial Differential Equations in the Plane Revisited
Author(s) -
Mshimba Ali Seif
Publication year - 1991
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.19911510126
Subject(s) - mathematics , sobolev space , mathematical analysis , boundary value problem , dirichlet problem , type (biology) , lipschitz domain , lipschitz continuity , elliptic boundary value problem , hilbert space , partial differential equation , pure mathematics , space (punctuation) , free boundary problem , ecology , linguistics , philosophy , biology
Both boundary value problems the DIRICHLET and the RIEMANN‐HILBERT problems, were solved by the author in the SOBOLEV space W 1,p (D) , 2 < p < ∞, for the elliptic differential equation in [8], [11] and [12]. In fact in most of the literature so far it is the case of p > 2 only which has been studied extensively. The results obtained so far relied on the fact that the VEKUA‐type operator T D maps the LEBESGUE space L p (D) Into the HÖLDER space ( D̄ ) for α= ( p –2)/ p . By modifying both VEKUA type singular integral operators T D and πD it is shown here that we can then solve both boundary value problems in W 1,p ( D ) for 1 < p < ∞. While enlarging the set of admissible SOBOLEV spaces, the set of partial differential equations that can then be solved is reduced through stricter conditions on the LIPSCHITZ constants.