By Michael Bildhauer
The writer emphasizes a non-uniform ellipticity situation because the major method of regularity thought for suggestions of convex variational issues of forms of non-standard progress conditions.
This quantity first makes a speciality of elliptic variational issues of linear progress stipulations. the following the idea of a "solution" isn't seen and the viewpoint needs to be replaced numerous occasions in an effort to get a few deeper perception. Then the smoothness houses of ideas to convex anisotropic variational issues of superlinear progress are studied. despite the basic variations, a non-uniform ellipticity situation serves because the major device in the direction of a unified view of the regularity conception for either types of problems.
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Additional resources for Convex Variational Problems: Linear, Nearly Linear and Anisotropic Growth Conditions
G The deﬁnition of σδ yields ∇f (∇uδ ) − ∇f (∇u∗ ) : (∇uδ − ∇u∗ ) η 2 dx G δ ∇uδ : ∇uδ − ∇u∗ η 2 dx + G (29) ∗ σδ : [uδ − u ] ⊗ ∇η η dx = −2 G ∇f (∇u∗ ) : [uδ − u∗ ] ⊗ ∇η η dx . +2 G Now, ∇f (∇u∗ ) ∈ L∞ (Ω; RnN ) and (uδ − u∗ ) → 0 in L1 (Ω; RnN ), hence ∇f (∇u∗ ) : [uδ − u∗ ] ⊗ ∇η η dx → 0 as δ → 0 . 3 Partial C 1,α - and C 0,α -regularity . . 2). The ﬁrst integral on the right-hand side of (29) can be written in the following form δ∇uδ : [uδ − u∗ ] ⊗ ∇η η dx −2 G ∇f (∇uδ ) : [uδ − u∗ ] ⊗ ∇η η dx =: I1 + I2 .
Moreover, the singular part ∇s u∗ is not necessarily vanishing on Ωcu∗ . 2. 10 are needed to assume without loss of generality (after passing to a subsequence) i) ii) σδ (x) → σ(x) for almost all x ∈ Ω , δ ∇uδ (x) → 0 for almost all x ∈ Ω , (35) 34 2 Variational problems with linear growth: the general setting where σδ = δ∇uδ + ∇f (∇uδ ) and where σ denotes the unique solution of the dual variational problem (P ∗ ). Passing to another subsequence, if necessary, a L1 -cluster point u∗ of uδ is ﬁxed in the following: L1 uδ −→: u∗ ∈ BV Ω, RN as δ → 0 .
Hence, u is a suitable candidate to solve (P) and ∇u is of class LF (Ω; RnN ). It remains to establish u ∈ KF : on account of (3) we obtain as above (using in addition (N3)) F |um | dx ≤ c , Ω where the constant is not depending on m. e. u ∈ WF1 (Ω; RN ) and u−u0 ∈ WF1 (Ω; RN )∩ W11 (Ω; RN ). 1 gives u ∈ KF , hence, u is a solution of the problem (P). 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 .