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

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.

Show description

Read or Download Convex Variational Problems: Linear, Nearly Linear and Anisotropic Growth Conditions PDF

Similar linear programming books

Combinatorial Data Analysis: Optimization by Dynamic Programming

Combinatorial information research (CDA) refers to a large type of equipment for the research of suitable info units within which the association of a suite of items is admittedly primary. the point of interest of this monograph is at the id of preparations, that are then extra constrained to the place the combinatorial seek is performed via a recursive optimization procedure according to the final ideas of dynamic programming (DP).

Science Sifting: Tools for Innovation in Science and Technology

Technology Sifting is designed basically as a textbook for college students attracted to examine and as a basic reference publication for current profession scientists. the purpose of this booklet is to aid budding scientists expand their capacities to entry and use details from different resources to the advantage of their study careers.

Additional resources for Convex Variational Problems: Linear, Nearly Linear and Anisotropic Growth Conditions

Example text

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 .

Download PDF sample

Rated 4.04 of 5 – based on 24 votes