By Władysław Orlicz
"Linear useful research" resulted from a chain of lectures Orlicz gave in Beijing, China, 1958. the unique variation used to be released in chinese language in 1963. It includes the entire significant theorems that will usually seem in a latest textual content, the result of exact curiosity to the Polish college, and others which aren't simply to be had in different places. Orlicz supplied during this ebook a few infrequent perception and motivation within the topic which used to be initiated by way of the Polish college. An addendum to a couple contemporary leads to Orlicz areas is incorporated.
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Additional resources for Linear functional analysis vol 4
1 and the Poincar`e inequality of [FO]. The uniqueness of solutions is immediate by the strict convexity of f . The existence and uniqueness theorem is derived by assuming just the growthcondition (2) for the strictly convex integrand f . From now on we concentrate on the regularity theory where we need some additional conditions on the second derivatives of f . These are made precise in the next sections. 4. Let F : [0, ∞) → [0, ∞) denote some N -function satisfying (N1)–(N3). Moreover, ﬁx some real number s ≥ 1 and assume that F (t) ≥ c0 ts for large values of t .
15). 17 will also serve as an important tool while proving the partial regularity results of the next chapter. In addition to R0 we now ﬁx radii r, R such that 0 < r < 2R ≤ 2R0 . 17. 11 without the restrictions (11) and (13). Moreover, let χ := n/(n − 2), if n ≥ 3. 11. 2−μ Then there are constants c ≡ c(f, r, R), β ≡ β(f ), independent of δ, such that: 1 + |∇uδ | 2 (2−μ)χ 2 β dx ≤ c Br (x0 ) 1 + fδ (∇uδ ) dx . 18. Note that our assumptions imply q < (2 − μ)χ. The proof given below in fact will show that in the two-dimensional case n = 2 we can choose χ as any ﬁnite number.
The density assumption (38) implies lim ϑ ρ→0 |K ∩ Bρ (ˆ x)| = ϑ 2−n , |B2ρ (ˆ x)| (47) whereas on account of (39), (40) and the boundedness of f∞ lim ρ→0 1 |B2ρ (ˆ x)| 1 + |B2ρ (ˆ x)| f (∇a u∗ ) dx B2ρ (ˆ x) f∞ B2ρ (ˆ x) ∇s u ∗ |∇s u∗ | (48) d|∇s u∗ | = 0. 29 is proved. 29 to the regularity theory: the u∗ - and the σ-degenerate sets are identiﬁed modulo sets of measure zero and, as a consequence, an intrinsic regularity theorem for σ is obtained for the degenerate problems under consideration.