By Catherine Bandle, Vitaly Moroz (auth.), Ari Laptev (eds.)
International Mathematical sequence quantity 12
Around the study of Vladimir Maz'ya II
Partial Differential Equations
Edited by way of Ari Laptev
Numerous influential contributions of Vladimir Maz'ya to PDEs are on the topic of different parts. particularly, the subsequent issues, just about the medical pursuits of V. Maz'ya are mentioned: semilinear elliptic equation with an exponential nonlinearity resolvents, eigenvalues, and eigenfunctions of elliptic operators in perturbed domain names, homogenization, asymptotics for the Laplace-Dirichlet equation in a perturbed polygonal area, the Navier-Stokes equation on Lipschitz domain names in Riemannian manifolds, nondegenerate quasilinear subelliptic equations of p-Laplacian kind, singular perturbations of elliptic platforms, elliptic inequalities on Riemannian manifolds, polynomial recommendations to the Dirichlet challenge, the 1st Neumann eigenvalues for a conformal classification of Riemannian metrics, the boundary regularity for quasilinear equations, the matter on a gentle stream over a two-dimensional challenge, the good posedness and asymptotics for the Stokes equation, indispensable equations for harmonic unmarried layer strength in domain names with cusps, the Stokes equations in a convex polyhedron, periodic scattering difficulties, the Neumann challenge for 4th order differential operators.
Contributors comprise: Catherine Bandle (Switzerland), Vitaly Moroz (UK), and Wolfgang Reichel (Germany); Gerassimos Barbatis (Greece), Victor I. Burenkov (Italy), and Pier Domenico Lamberti (Italy); Grigori Chechkin (Russia); Monique Dauge (France), Sebastien Tordeux (France), and Gregory Vial (France); Martin Dindos (UK); Andras Domokos (USA) and Juan J. Manfredi (USA); Yuri V. Egorov (France), Nicolas Meunier (France), and Evariste Sanchez-Palencia (France); Alexander Grigor'yan (Germany) and Vladimir A. Kondratiev (Russia); Dmitry Khavinson (USA) and Nikos Stylianopoulos (Cyprus); Gerasim Kokarev (UK) and Nikolai Nadirashvili (France); Vitali Liskevich (UK) and Igor I. Skrypnik (Ukraine); Oleg Motygin (Russia) and Nikolay Kuznetsov (Russia); Grigory P. Panasenko (France) and Ruxandra Stavre (Romania); Sergei V. Poborchi (Russia); Jurgen Rossmann (Germany); Gunther Schmidt (Germany); Gregory C. Verchota (USA).
Imperial collage London (UK) and
Royal Institute of know-how (Sweden)
Ari Laptev is a world-recognized expert in Spectral idea of
Differential Operators. he's the President of the ecu Mathematical
Society for the interval 2007- 2010.
Sobolev Institute of arithmetic SB RAS (Russia)
and an self sufficient publisher
Editors and Authors are solely invited to give a contribution to volumes highlighting
recent advances in a number of fields of arithmetic by way of the sequence Editor and a founder
of the IMS Tamara Rozhkovskaya.
Cover snapshot: Vladimir Maz'ya
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Additional info for Around the Research of Vladimir Maz'ya II: Partial Differential Equations
Barbatis et al. 9) without any smallness assumptions on δ∞ (φ, φ) and δpr (φ, φ) respectively. 8). 1). It is clear that λn [H] 1+ |λn [H] − ξ| |ξ| . 11) Since the eigenvalues of the operator wT ∗ ST w coincide with the eigenvalues of H (cf. 2)), it follows that ∞ (wT ∗ ST w − ξ)−1 1 + |ξ|α = 1+ α Cα 1 = n=1 α |ξ| |λn [H] − ξ|α λn [H]−α d(ξ, σ(H)) 1 |ξ| +c 1+ α |ξ| d(ξ, σ(H)) λn [H]=0 α . N ∇φ − ∇φ max |∇φ| , ∇φ |1 − w|, |w − w−1 | N −1 , c|∇φ − ∇φ|. 14), we obtain A1 Cα , A2 A3 Cα c |ξ| 1 + ∇φ − ∇φ L∞ (Ω) , |ξ| dσ (ξ) 1 + |ξ| |ξ|2 + ∇φ − ∇φ L∞ (Ω) .
87, 37–56 (2007) 7. : On solutions of ∆u = f (u). Commun. Pure Appl. Math. 10, 503–510 (1957) 8. : On the best constant for Hardy’s inequality in Rn . Trans. Am. Math. Soc. 350, 3237–3255 (1998) 9. : Sobolev Spaces. Springer, Berlin etc. (1985) 10. : On the inequality ∆u f (u). Pacific J. Math. 7, 1641–1647 (1957) 11. : Singular Integrals and Differentiability Properties of Functions. Princeton Univ. Press, Princeton, NJ (1970) Stability Estimates for Resolvents, Eigenvalues, and Eigenfunctions of Elliptic Operators on Variable Domains Gerassimos Barbatis, Victor I.
12]). In the next lemma, g(H) and g(F ) are operators defined in the standard way by functional calculus. The following lemma is a variant of Lemma 8 of . 5. Let q0 > 2, γ 0, p q0 /(q0 − 2), 2 2q0 γ/[p(q0 − 2)]. Then the following statements hold. r < ∞ and s = (i) If the eigenfunctions of H satisfy (P1), then for any measurable function R : Ω → C and function g : σ(H) → C we have Rg(H) R Cr 2q0 1 Lpr (Ω) |Ω|− pr |g(0)| + C pr(q0 −2) |g(H)|r,s . 5) (ii) If the eigenfunctions of H satisfy (P2), then for any measurable matrixvalued function R in Ω and function g : σ(F ) → C such that if 0 ∈ σ(F ), then g(0) = 0, we have Rg(F ) 2q0 Cr C pr(q0 −2) a 1 r L∞ (Ω) R Lpr (Ω) |g(F )|r,s .