David E. Edmunds

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Let m and n be positive integers with n 2 and 1 m n ? 1. We study rearrangement-invariant quasinorms % R and % D on functions f : (0; 1) ! R such that to each bounded domain in R n , with Lebesgue measure jj, there corresponds C = C(jj) > 0 for which one has the Sobolev imbedding inequality % R ? u (jjt) C% D ? jr m uj (jjt) ; u 2 C m 0 ((); involving the(More)
Some three-dimensional analogues of the plane Darboux problems for hyperbolic equations with degeneracy are investigated. In 1954, Protter initiated the study of such three-dimensional problems, and it is now well known that for an infinite number of smooth right-hand sides these problems have solutions with a strong power-type singularity on the(More)