By Mariano Giaquinta

This quantity offers with the regularity conception for elliptic structures. We may possibly locate the starting place of this sort of thought in of the issues posed through David Hilbert in his celebrated lecture brought throughout the foreign Congress of Mathematicians in 1900 in Paris: nineteenth challenge: Are the strategies to standard difficulties within the Calculus of adaptations continuously unavoidably analytic? twentieth challenge: does any variational challenge have an answer, only if definite assumptions in regards to the given boundary stipulations are happy, and only if the inspiration of an answer is definitely prolonged? over the past century those difficulties have generated loads of paintings, often known as regularity concept, which makes this subject particularly appropriate in lots of fields and nonetheless very energetic for examine. besides the fact that, the aim of this quantity, addressed frequently to scholars, is far extra restricted. 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. actually a few proper themes are passed over. issues contain: harmonic services, direct equipment, Hilbert house tools and Sobolev areas, power estimates, Schauder and L^p-theory either with and with no capability conception, 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 bigger than 1. during this moment deeply revised version we additionally integrated 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 thought, together with the prestigious result of Wente and of Coifman-Lions-Meyer-Semmes.

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

**Sample text**

BV -functions are exactly the functions having graphs of ﬁnite area. In particular, given any u ∈ BV (Ω) or any u ∈ C 0 (Ω) with ﬁnite area, there exists a sequence uk ∈ C ∞ (Ω) (C ∞ (Ω) provided, of course ∂Ω is smooth) such that uk → u in L1 (or uniformly) and F(uk ) → F(u). g. [6] [49] [51]. 1 BV minimizers for the area functional We now want to use direct methods to prove existence of minimal graphs with prescribed boundary. The natural space to work with is BV (Ω). Since a function u ∈ L1 (Ω) is deﬁned up to a set of zero measure, we cannot na¨ıvely make sense of the boundary datum u ∂Ω .

Proof. Step 1. Deﬁne on the Hilbert space H := W01,2 (Ω) the bilinear form Aαβ Dα vDβ wdx, B(v, w) := Ω which is coercive thanks the Poincar´e inequality and the ellipticity of Aαβ . Set u = u − g, so that the initial problem is reduced to ﬁnding u ∈ H such that for every v ∈ W01,2 (Ω) Ω Aαβ Dα uDβ vdx = Ω f0 vdx + Ω [Aαβ Dα g + f β ]Dβ vdx =: L(v). 30. 11). 3 Elliptic equations: existence of weak solutions 51 Step 2. If Aαβ is symmetric, the second derivative of F is D2 Fu (v, w) = Ω Aαβ Dα vDβ wdx, so that F is convex on W 1,2 and strictly convex on A.

Then, for 1 ≤ p < +∞, the following immersion W 1,p (Ω) → Lp (Ω) is compact. Proof. We ﬁrst show that the immersion W 1,p (Q) → Lp (Q) is compact, where Q is a cube of side . Let {uk } ⊂ W 1,p (Q) with uk W 1,p ≤ M. Fix ε > 0 and let Q1 , . . , Qs be a subdivision of Q in cubes with disjoint interiors and side σ, σ < ε. Of course |(uk )Qj | := Qj uk (x)dx ≤ c . σn 44 Hilbert space methods Consider the ﬁnite family G of simple functions g(x) = n1 εχQ1 (x) + . . + ns εχQs (x), where n1 , . . , ns are integers in (−N, N ) with N > εσcn and χQj is the characteristic function of Qj .