Convex Variational Problems: Linear, Nearly Linear and by Michael Bildhauer

By Michael Bildhauer

The writer emphasizes a non-uniform ellipticity because the major method of regularity idea for ideas of convex variational issues of kinds of non-standard progress conditions.

This quantity first makes a speciality of elliptic variational issues of linear development stipulations. right here the concept of a "solution" isn't really noticeable and the perspective needs to be replaced numerous instances so one can get a few deeper perception. Then the smoothness houses of suggestions to convex anisotropic variational issues of superlinear development are studied. inspite of the elemental ameliorations, a non-uniform ellipticity situation serves because the major device in the direction of a unified view of the regularity idea for either varieties of problems.

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Extra info for Convex Variational Problems: Linear, Nearly Linear and Anisotropic Growth Conditions

Example text

15 also covers the degenerate case in the sense that D2 f (Z)(Y, Y ) = 0 is not excluded. 13 below is the main tool to prove the uniqueness of the dual solution. Here and in the following we let U := Im(∇f ). 13. 2 satisfies x ∈ Ω : σ(x) ∈ ∂U = 0. Here | · | denotes the Lebesgue measure Ln . For the proof we need the following observation. 14. For all real numbers K > 0 there is an ε > 0 such that for all Z ∈ RnN dist ∇f (Z), ∂U < ε ⇒ |Z| > K . 14. e. ∇f is a one-to-one mapping. In fact, we have ∇f (Z) − ∇f (Y ) : (Z − Y ) 1 D2 f sZ + (1 − s)Y = (Z − Y ), (Z − Y ) ds ≥ 0 .

17. 1. Moreover, consider a J-minimizing sequence {um } from the affine class ◦ u0 + W11 (Ω; RN ) and u∗ ∈ L1 (Ω; RN ) satisfying um → u∗ in L1 (Ω; RN ) as m → ∞ . If Ωu∗ is given according to (23) then Ωu∗ is an open set and we have u∗ ∈ C 1,α (Ωu∗ ; RN ) for any α ∈ (0, 1) . 18. 1 0 < D2 f (Z)(Y, Y ) for all Z, Y ∈ RnN , Y = 0 . e. |Ω − Ω0 | = 0, such that u∗ ∈ C 1,α (Ω0 ; RN ) for any α ∈ (0, 1) . 1 for a more intensive discussion). 16). ˆ always denotes a bounded Lipschitz domain. 19. For all w ∈ BV (Ω; ˆ RN ), wk → w in L1loc (Ω; ˆ RN ) ˆ Ω] ˆ := inf lim inf J[wk ] : wk ∈ C 1 (Ω; J[w; k→∞ .

This is evident if we consider Y parallel to Z. The second inequality of iii) again is immediate. Next we are going to prove iv): observing 1 |D2 Φ(Z)| = sup D2 Φ(Z)(Y, Y ) ≤ 2 |Y |=1 1 + t2 |Z|2 −μ 2 dt 0 we get |Z| |D2 Φ(Z)| |Z|2 ≤ 2 |Z| 1 + s2 −μ 2 ∞ ds ≤ 2 |Z| 0 1 + s2 −μ 2 ds , 0 the last integral being finite on account of μ > 1. 10. 9 provides a regular class of integrands with linear growth (with some appropriate choice of μ > 1). This will be proved in the next chapter. 17 for a further discussion of Φ in the case μ ≥ 1.

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