By Giuseppe Mingione
The Willmore sensible / Ernst Kuwert and Reiner Schatzle -- The function of conservation legislation within the research of conformally invariant difficulties / Tristan RivieМЂre -- Regularity homes of equilibrium configurations of epitaxially strained elastic motion pictures / Bruno De Maria and Nicola Fusco
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Extra info for Topics in modern regularity theory
This was rigorously proved in [Bl09]. Still it is open whether the singularity develops in ﬁnite time or at inﬁnity. 3 Blow up To study the singulartities in greater detail, we set up a blow up procedure. With the Michael-Simon-Sobolev inequality and the absorption technique in the above theorem, one can prove interior estimate stated in the next theorem. 5). 1) B (0) for some > 0 and appropriate ε1 (n) > 0. Then we have for 0 < t < c0 (n) 4 , T and for any k ∈ N0 √ ∇ ⊥,k At L ∞ (B1/2 (0)) ≤ Ck (n) ε t −(k+1)/4 , √ ∇ ⊥,k At L 2 (B1/2 (0)) ≤ Ck (n) ε t −k/4 .
For appropriate ε0 = ε0 (n) > 0, we consider > 0 satisfying |A0 |2 dμ f0 ≤ ε ≤ ε0 for all x ∈ Rn . 3) B (x) for 0 < c0 , C < ∞ depending only on n. Sketch of the Proof. To abbreviate notation for normal tensors φ, ψ, we denote by φ ∗ ψ any normal tensor obtained by contraction in a bilinear way, and we write ∇ ⊥,i1 A ∗ ∇ ⊥,i2 A ∗ . . ∗ ∇ ⊥,ir A. +ir =m Clearly, m+l (A) and ∇ ⊥ Prm (A) = Prm+1 (A). 4) that ∂t⊥ Ai j = ∇i⊥ ∇ ⊥j V − A jk , V g kl Ail = ∇i⊥ ∇ ⊥j V + A ∗ A ∗ V. 4) H + P32 (A) + P50 (A).
6]). Let f : → Rn be a proper Willmore immersion of an open surface satisfying |A0f |2 dμ f ≤ ε1 , lim −4 →∞ in particular if B (0) |A f |2 dμ f = 0, |A f |2 dμ f < ∞, then (|∇ ⊥,2 A|2 + |A|2 |∇ ⊥ A|2 + |A|4 |A0 |2 ) dμ f ≤ Cn |δW( f )|2 dμ f . This proposition implies the following gap lemma. 7]). Under the assumptions of the previous proposition, if f is Willmore, then f parametrises on each connected component of an embedded plane or a round sphere. Proof. f is Willmore if and only if δW( f ) = 0, hence A0 ≡ 0, and f maps each connected component of into an embedded plane or a round sphere by Codazzi’s theorem, say f : 0 → M for 0 ⊆ connected componenet and M = P a plane or = S 2 .