@MISC{(firenze_bestconstant, author = {Giorgio Talenti (firenze}, title = {Best Constant in Sobolev Inequality (*).}, year = {} }

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Abstract

Summary.- The best constant Jot the simplest Sobolev inequality is exhibited. The proo] is accomplished by symmetrizations (rearrangements in the sense o] Hardy-Littlewood) and one-dimensional calculus o] variations. 0.- The main result of the present paper is the following: THnom~t.- Let u be any real (or complex) valued]unction, de]ined on the whole m-dimensional euclidean space R ~ su]]iciently smooth and decaying Just enough at inJinity. Moreover let p be any number such that: l < p < m. Then: (1) { fluIcdx}llq<C { flDul~dx} 1I ~, tgm- _ 1¢m-where: IDul is the length o] the gradient Duo] u, q- = m]~/(m--p) and (2) c Ira-p) |r(m/p)r(1 + m-m/p) 7~-~m- ~............. The equality sign holds in (1) i] u has the Jorm: (3) u(x) = [a + btxI,~',-'] 1-~1 ~, where Ix [ = (x ~ @...-~- x~) and a, b are positive constants. Sobolev inequalities, also called Sobolev imbedding theorems, are very popular