# Download PDF by Verena Bogelein, Frank Duzaar, Giuseppe Mingione: The regularity of general parabolic systems with degenerate

By Verena Bogelein, Frank Duzaar, Giuseppe Mingione

ISBN-10: 0821889753

ISBN-13: 9780821889756

The purpose of the paper is twofold. On one hand the authors are looking to current a brand new procedure known as $p$-caloric approximation, that's a formal generalization of the classical compactness tools first constructed by means of DeGiorgi together with his Harmonic Approximation Lemma. This final end result, at the beginning brought within the surroundings of Geometric degree concept to end up the regularity of minimum surfaces, is these days a classical device to turn out linearization and regularity effects for vectorial difficulties. the following the authors increase a truly a long way attaining model of this basic precept devised to linearize normal degenerate parabolic structures. using this lead to flip permits the authors to accomplish the next and major target of the paper, that's, the implementation of a partial regularity thought for parabolic platforms with degenerate diffusion of the sort $\partial_t u - \mathrm{div} a(Du)=0$, with no unavoidably assuming a quasi-diagonal constitution, i.e. a constitution prescribing that the gradient non-linearities count in simple terms at the the specific scalar volume

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Additional resources for The regularity of general parabolic systems with degenerate diffusion

Example text

M−1} such that gk dz ≤ ε. i(k) Fk Since the map N k → i(k) ∈ {0, . . , m − 1} takes values in the ﬁnite set {0, . . , m − 1} there exists at least one ∈ {0, . . , m − 1} and an inﬁnite subset K ⊆ N such that i(k) = for any k ∈ K, and choosing λ = λ20 yields the assertion of the lemma. 1. 2 dealing with the case p ≥ 2. 2. e. for the case z0 = 0 and = 1 since we can always come in this situation by a scaling argument. Indeed, we can rescale w to Q1 via W (x, t) ≡ −1 w(x0 + x, t0 + 2 t) to obtain the existence of an p-caloric map H on Q1/2 .

2n (λ) With regard to the time direction a similar argument applies. 2. LIPSCHITZ TRUNCATION 35 obtain |Λ3/4 ∩ Λ(λ) (t0 )| ≥ λ2−p 2 ≥ 16−2 λ2−p λ− n+2 = 16−2 λ n+2 −p . 4 2n 2n Here we also used that t0 ∈ Λ3/4 . Since p ≥ n+2 and λ ≥ 1 we have in any case that there holds 2n |Λ3/4 ∩ Λ(λ) (t0 )| ≥ 16−2 λ n+2 −p . Joining the preceding estimates we arrive at |Q3/4 ∩ Q(λ) (z0 )| ≥ αn 32−(n+2) λ−p . 7) in the deﬁnition of Iz0 ( ) we obtain p I (z0 ) ≤ c λp+ n+2 v λ p L1 (Q3/4 ) ≤ c(n) λp+ n+2 v L1 (Q1 ) .

Separating the cylinders into these groups we obtain ∂t [v λ ]h η · [v − v λ ]h dz Q1 ≤ ∂t [v λ ]h η i∈Nh Qi −h |v − v λ | dz + ∂t [v λ ]h η i∈N\Nh Qi −h |v − v λ | dz. e. i ∈ Nh . Using that 3 λ2−p ri2 + h ≤ 2λ2−p ri2 ≤ λ2−p ( ri )2 2 for i ∈ Nh and the choices of spt η ⊆ Q5/8 and h ≤ 1/8 we ﬁnd sup ∂t [v λ ]h η Qi −h ≤ η L∞ sup Q3/4 ∩ 32 Qi |∂t v λ | ≤ c λp−1 ri 1 + λp v L1 (Q1 ) , where c = c(n) η L∞ . 24) for |∂t v λ |. 6). Joining the last two estimates and keeping in mind that rj ≤ 3ri for j ∈ I(i) we infer that ∂t [v λ ]h η Qi ≤ c(n) η −h |v − v λ | dz L∞ λ 1 + λp v λp−2 |f | + |f |p−1 dz, L1 (Q1 ) j∈I(i) 2Qj whenever i ∈ Nh .