Geometric Applications of Fourier Series and Spherical by Helmut Groemer
By Helmut Groemer
This can be the 1st accomplished exposition of the appliance of round harmonics to turn out geometric effects. the writer offers the entire worthy instruments from classical conception of round harmonics with complete proofs. Groemer makes use of those instruments to end up geometric inequalities, strong point effects for projections and intersection via planes or half-spaces, balance effects, and characterizations of convex our bodies of a selected style, resembling rotors in convex polytopes. effects coming up from those analytical concepts have proved precious in lots of purposes, relatively these regarding stereology. To make the remedy as self-contained as attainable the booklet starts with heritage fabric in research and the geometry of convex units.
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Sample text
Pisa (2004) 16. : On the Navier-Stokes initial value problem I. Arch. Rat. Mech. Anal. 16, 269–315 (1964) 17. : Strong L p -solutions of the Navier-Stokes equations in Rm with applications to weak solutions. Math. Z. 187, 471–480 (1984) 18. : Solutions autosimilaires des équations de Navier-Stokes, Séminaire Équations aux Dérivées Partielles de l’École Polytechnique, 1993–1994 19. : Théorémes d’unicité pour le systéme de Navier-Stokes tridimensionnel. J. Anal. Math. 77, 27–50 (1999) 20. : Well-posedness for the Navier-Stokes equations.
Ii) Let s ∈ R, 1 ≤ p, r ≤ ∞. Then u belongs to B sp,r if and only if there exists {c j,r } j∈N such that c j,r r = 1 and ˙ ju Lp ≤ Cc j,r 2− js u B˙ sp,r . In order to obtain a better description of the regularizing effect of the transportdiffusion equation, we will use Chemin-Lerner type spaces L λT ( B˙ sp,r (R3 )) from [19, 39]. 3 Let s ≤ 3p (resp. s ∈ R), (r, λ, p) ∈ [1, +∞]3 and T ∈]0, +∞]. We define L λT ( B˙ sp r (R3 )) as the completion of C([0, T ], S(R3 )) by the norm f L λT ( B˙ sp,r ) def = T 2qr s q∈Z 0 ˙ q f (t) λ Lp dt r λ 1 r < ∞.
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