By Michael Bildhauer
The writer emphasizes a non-uniform ellipticity situation because the major method of regularity idea for recommendations of convex variational issues of varieties of non-standard development conditions.
This quantity first specializes in elliptic variational issues of linear progress stipulations. right here the suggestion of a "solution" isn't really visible and the viewpoint should be replaced numerous occasions as a way to get a few deeper perception. Then the smoothness homes of options to convex anisotropic variational issues of superlinear development are studied. despite the basic transformations, a non-uniform ellipticity situation serves because the major software in the direction of a unified view of the regularity concept for either sorts of problems.
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Extra resources for Convex Variational Problems: Linear, Nearly Linear and Anisotropic Growth Conditions
2 we could try to add a δ-term of power t > q instead of the above choice. This, however, would yield some serious technical diﬃculties in the scalar case (and for vectorial problems with additional structure), where it is not evident how to handle higher order δ-terms in order to make the DeGiorgi technique work. 4, the choice t = q gives some diﬃculties concerning the starting integrability since we cannot refer to the discussion of asymptotic regular integrands as given in [CE] or [GiaM]. 17.
Note that this implies ∂2f ∂zαi ∂zβj |Z|2 zαi zβj + 2 g |Z|2 δ ij δαβ , (Z) = 4 g (12) α, β = 1, . . , n; i, j = 1, . . , N . We also assume in the case N > 1 that there are real numbers α ∈ (0, 1], K > 0 satisfying for all Z, Z˜ ∈ RnN ˜ ≤ K |Z − Z| ˜α. D2 f (Z) − D2 f (Z) (13) Note that the above examples are easily adjusted to this conditions (in fact to much stronger ones) by letting √ ε+r 2 s ϕ(r) = 0 0 1 + t2 −μ 2 dt ds , ε>0. 11. Given μ ∈ (−∞, 2), 1 ≤ s ≤ q, 1 < q, assume that f is of (s, μ, q)-growth.
We proceed in three steps. Step 1. Let q > 1 and start with the q-power growth integrand ρ(t) = (1 + t2 )q/2 . Then we “destroy” ellipticity by deﬁning (for all n ∈ N0 ) ρ˜(t) = ρ(t) if 2n ≤ t < 2n + 1, whereas for 2n + 1 ≤ t < 2n + 2 we let ρ˜(t) = ρ(2n + 1) + t − (2n + 1) ρ(2n + 2) − ρ(2n + 1) . Moreover, the function ρ˜ is extended to the whole line by setting ρ˜(−t) = ρ˜(t). 6. There is a positive constant c such that i) (˜ ρ)ε is a (smooth) N-function. ii) Let g(Z) = (˜ ρ)ε |Z| , Z ∈ RnN .
Convex Variational Problems: Linear, Nearly Linear and Anisotropic Growth Conditions by Michael Bildhauer