Download Perspectives in Partial Differential Equations, Harmonic by Dorina Mitrea, Marius Mitrea PDF

By Dorina Mitrea, Marius Mitrea

V. G. Mazya is broadly considered as a very extraordinary mathematician, whose paintings spans 50 years and covers many parts of mathematical research. This quantity features a exact selection of papers contributed at the get together of Mazya's seventieth birthday by way of a distinct team of specialists of foreign stature within the fields of Harmonic research, Partial Differential Equations, functionality idea, Spectral research, and historical past of arithmetic, reflecting the cutting-edge in those components, during which Mazya himself has made a few of his most important contributions.

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Proceeding by induction on s, we obtain thus (s1 ) 2 |a(0) n,n1 · · · ans ,m | I 1 m∈Zd n1 ,... ,ns1 1/2 ns1 +2,... 12). 2 This proves Lemma 1. Expressions considered in Lemma 1 appear when writing out matrix elements of products (A(0) Vω A(1) Vω · · · A(s) )(n, n ) where the A(s ) are matrices and Vω a random potential Vω (n) = ωn vn . 16) (A(0) Vω A(1) Vω · · · A(s) )(∗) to indicate that, when writing out the matrix product as a sum over multi-indices, we do restrict the sum to the admissible multi indices.

L2 This raises the question Problem 4. Compute inf Ω AΩ and inf Ω SΩ . Are they achieved? 1], the question was asked whether inf Ω SΩ = SB(0,1) .

However, it was shown recently that in circulation integrals, such functions behave as if they were bounded continuous functions: Theorem 1 (Bourgain, Brezis and Mironescu [4]). 1) Ln (Rn ) |Γ| , Γ 1991 Mathematics Subject Classification. 26D10, 35J25, 46E35, 49Q15, 51M16, 58C35. Key words and phrases. Circulation integrals, optimal constants, Sobolev inequalities, isoperimetric inequalities, divergence-free vector fields. The second author (JVS) was partially supported by the Fonds de la Recherche Scientifique– FNRS (Belgium) and by the Fonds Sp´ecial de Recherche (Universit´e catholique de Louvain).

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