By Michael Bildhauer
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Additional resources for Convex Variational Problems: Linear, Nearly Linear and Anisotropic Growth Conditions
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.