By Mariano Giaquinta

This quantity bargains with the regularity idea for elliptic platforms. We might locate the foundation of this sort of idea in of the issues posed by means of David Hilbert in his celebrated lecture added through the overseas Congress of Mathematicians in 1900 in Paris: nineteenth challenge: Are the options to standard difficulties within the Calculus of diversifications consistently unavoidably analytic? twentieth challenge: does any variational challenge have an answer, only if definite assumptions concerning the given boundary stipulations are happy, and only if the concept of an answer is definitely prolonged? over the last century those difficulties have generated loads of paintings, often known as regularity idea, which makes this subject relatively suitable in lots of fields and nonetheless very energetic for study. in spite of the fact that, the aim of this quantity, addressed normally to scholars, is way extra constrained. We target to demonstrate just some of the elemental principles and strategies brought during this context, confining ourselves to big yet basic occasions and refraining from completeness. in reality a few suitable themes are passed over. themes comprise: harmonic services, direct tools, Hilbert area tools and Sobolev areas, strength estimates, Schauder and L^p-theory either with and with out power thought, 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; strength minimizing harmonic maps and minimum graphs in codimension 1 and larger than 1. during this moment deeply revised version 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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**Additional resources for An Introduction to the Regularity Theory for Elliptic Systems, Harmonic Maps and Minimal Graphs**

**Example text**

In order to understand which functions have ﬁnite relaxed area, we extend the above deﬁnition to L1 , replacing the uniform convergence with the L1 convergence: for each u ∈ L1 (Ω) F(u) = inf lim inf F(uk ) : uk → u in L1 , uk ∈ C 1 (Ω) . 29 An L1 (Ω) function is said to be of bounded variation when its partial derivatives in the sense of distributions are signed measures with ﬁnite total variation. The subspace of L1 (Ω) consisting of such functions is called BV (Ω). 12) where n F(u) := sup u Ω i=1 Di gi + gn+1 dx : g ∈ Cc1 (Ω, Rn+1 ), |g| ≤ 1 .

2 Sharpness of the mean curvature condition We now show that, at least in the case of the area functional F(u) = condition (ii)b is sharp. 20 Let x0 ∈ ∂Ω be such that H(x0 ) < 0, where it is assumed that Ω is a C 2 domain. e. the area functional F has no minimizer in A = {u ∈ Lip(Ω) : u = g on ∂Ω}. 22 below by choosing ε > 0, consequently ﬁxing Γ and ﬁnally imposing g = 0 on ∂Ω\Γ and g(x0 ) > 2ε . 22 we will need the following lemma. 21 Let u ∈ Lip(Ω) be a subsolution and a supersolution of F and let v ∈ C 1 (Ω) ∩ C 0 (Ω) be a supersolution of F.

We now show that each uk has Lp -distance from one of the g in G not greater than c2 ε for some c2 ( , n, p): this concludes the proof since then G is a ﬁnite c2 ε-net. Deﬁne s u∗k (x) := (uk )Qj χQj (x). 2) yields Q |uk − u∗k |p dx ≤ s j=1 Qj |uk − u∗k |p dx s ≤ c(n, p)σ p j=1 Qj |Duk |p ≤ cM p σ p . On the other hand, there is g ∈ G such that for all x ∈ Q |g(x) − u∗k (x)| < ε, hence uk − g Lp ≤ uk − u∗k Lp + u∗k − g Lp ≤ c1 σ + n ε ≤ c2 ε. In order to complete the proof, use the extension property to extend (with uniform bounds on the norms) every function in W 1,p (Ω) to a function in W 1,p (Q) for a cube Q ⊃⊃ Ω and then apply the previous part of the proof.