By Mariano Giaquinta

This quantity bargains with the regularity concept for elliptic platforms. We could locate the foundation of the sort of concept in of the issues posed by means of David Hilbert in his celebrated lecture brought throughout the overseas Congress of Mathematicians in 1900 in Paris: nineteenth challenge: Are the strategies to average difficulties within the Calculus of diversifications regularly unavoidably analytic? twentieth challenge: does any variational challenge have an answer, only if definite assumptions in regards to the given boundary stipulations are chuffed, and only if the inspiration of an answer is certainly prolonged? over the last century those difficulties have generated loads of paintings, often often called regularity concept, which makes this subject relatively suitable in lots of fields and nonetheless very energetic for examine. despite the fact that, the aim of this quantity, addressed ordinarily to scholars, is far extra constrained. We objective to demonstrate just some of the fundamental rules and strategies brought during this context, confining ourselves to big yet basic occasions and refraining from completeness. in truth a few proper subject matters are passed over. themes contain: harmonic services, direct tools, Hilbert house tools and Sobolev areas, power estimates, Schauder and L^p-theory either with and with no capability concept, together with the Calderon-Zygmund theorem, Harnack's and De Giorgi-Moser-Nash theorems within the scalar case and partial regularity theorems within the vector valued case; power minimizing harmonic maps and minimum graphs in codimension 1 and larger than 1. during this moment deeply revised variation we additionally incorporated the regularity of 2-dimensional weakly harmonic maps, the partial regularity of desk bound harmonic maps, and their connections with the case p=1 of the L^p conception, together with the prestigious result of Wente and of Coifman-Lions-Meyer-Semmes.

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**Extra resources for An Introduction to the Regularity Theory for Elliptic Systems, Harmonic Maps and Minimal Graphs**

**Example text**

13 The BSC is a pretty strong condition: for instance, it can be true only if Ω is convex. On the other hand, notice that the above result holds for a wide class of functionals. 3 Constructing barriers: the distance function Since the BSC is very restrictive, we will discuss other conditions on a domain Ω and a function g ∈ Lip(∂Ω) which allow to construct barriers and minimize a given variational integral F(u) = Ω F (Du)dx, with F convex and Fp continuous. 14 Given a boundary datum g ∈ Lip(∂Ω), an upper barrier at x0 ∈ ∂Ω is a supersolution b+ ∈ Lip(Ω) of F such that b+ (x0 ) = g(x0 ) and b+ ≥ g on ∂Ω.

3 Constructing barriers: the distance function Since the BSC is very restrictive, we will discuss other conditions on a domain Ω and a function g ∈ Lip(∂Ω) which allow to construct barriers and minimize a given variational integral F(u) = Ω F (Du)dx, with F convex and Fp continuous. 14 Given a boundary datum g ∈ Lip(∂Ω), an upper barrier at x0 ∈ ∂Ω is a supersolution b+ ∈ Lip(Ω) of F such that b+ (x0 ) = g(x0 ) and b+ ≥ g on ∂Ω. Lower barriers are deﬁned analogously. 24 Direct methods Suppose that Ω is of class C k , k ≥ 1; then there exist an interior tubular neighborhood N of ∂Ω, N = {x ∈ Ω : dist(x, ∂Ω) < ε}, where the corresponging projection p : N → ∂Ω is of class C k−1 .

By Rellich’s and Banach-Alaoglu’s theorems we may ﬁnd a subsequence (unk ) such that Lp unk → u, unk W 1,p u. e. u is constant, u Lp (Ω) = 1 and uΩ = 0, which is clearly impossible. The last claim of the proposition follows by scaling. 4 The chain rule in Sobolev spaces The following properties of Sobolev functions are often used. As usual we will consider Ω bounded. 22 Let f ∈ C 1 (R) with f ∈ L∞ (R) and u ∈ W 1,p (Ω) for some p ∈ [1, ∞]. Then f ◦ u ∈ W 1,p (Ω) and D(f ◦ u) = f (u)Du. Proof. It clearly suﬃces to prove the proposition for p = 1, since u ∈ W 1,p (Ω) implies u ∈ W 1,1 (Ω), hence (by the case p = 1) the weak derivative of f ◦u is f (u)Du which clearly belongs to Lp , hence f ◦u ∈ W 1,p (Ω).