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Boundary value problems in weighted spaces. (English) Zbl 0627.46032
Differential equations and their applications, Equadiff 6, Proc. 6th Int. Conf., Brno/Czech. 1985, Lect. Notes Math. 1192, 35-48 (1986).
[For the entire collection see Zbl 0595.00009.]
This paper is a survey of some recent work by the author (and some co- workers) concerning the use of weighted Sobolev spaces to solve partial differential equations involving linear differential operators of order 2k. Different situations where the usual approach by using Sobolv spaces \(W^{k,2}(D)\) does not work are considered: singular or degenerate coefficients, “bad” right-hand sides, non-smooth boundary data, etc. The main tool for proving existence and uniqueness of a weak solution is still the Lax-Milgram Lemma or a variant of it due to Nečas. A nonlinear version of these results is also included. Many extensions and variants are listed together with the corresponding literature.
Reviewer: J.Hernandez

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
35J35 Variational methods for higher-order elliptic equations
35J65 Nonlinear boundary value problems for linear elliptic equations
47J05 Equations involving nonlinear operators (general)
35R05 PDEs with low regular coefficients and/or low regular data
Zbl 0595.00009
Full Text: EuDML