By Mariano Giaquinta
This quantity offers with the regularity conception for elliptic structures. We might locate the starting place of this kind of thought in of the issues posed by way of David Hilbert in his celebrated lecture added in the course of the foreign Congress of Mathematicians in 1900 in Paris: nineteenth challenge: Are the suggestions to general difficulties within the Calculus of diversifications constantly 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 thought of an answer is certainly prolonged? over the past century those difficulties have generated loads of paintings, often known as regularity idea, which makes this subject relatively correct in lots of fields and nonetheless very energetic for examine. besides the fact that, the aim of this quantity, addressed regularly to scholars, is far extra restricted. We target to demonstrate just some of the elemental rules and methods brought during this context, confining ourselves to big yet basic occasions and refraining from completeness. in truth a few appropriate subject matters are passed over. subject matters comprise: harmonic capabilities, direct equipment, Hilbert house 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 bigger 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 idea, 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
23. 25 Given u ∈ W 1,p (Ω), 1 ≤ p ≤ ∞ and k ∈ R we have (u − k)+ ∈ W 1,p (Ω) and Du(x) if u(x) > k 0 if u(x) ≤ k. D(u − k)+ (x) = Proof. 5 t−k 0 if t ≥ k if t ≤ k. 6) and let p ∈ [1, +∞), k ≥ 1. Then 1. 7) 2. 3 Elliptic equations: existence of weak solutions 49 3. 9) for every u ∈ W 1,p (Ω). 9) is due to Morrey . 27 For every bounded and connected domain Ω with the extension property there is a constant c = c(p, Ω) such that for every u ∈ W 1,p (Ω), 1 ≤ p < ∞, we have 1 p∗ p∗ Ω |u − uΩ | dx ≤c p Ω |Du| dx 1 p , where uΩ := –Ω udx.
36 A matrix of coeﬃcients Aαβ ij 1≤α,β≤n 1≤i,j≤m is said to satisfy 1. 16) 2. the strong ellipticity condition, or the Legendre-Hadamard condition, if there is a λ > 0 such that i j 2 2 Aαβ ij ξα ξβ η η ≥ λ|ξ| |η| , ∀ξ ∈ Rn , ∀η ∈ Rm . 16). The converse is trivially true in case m = 1 or n = 1, but is false in general as the following example shows. 38 Let n = m = 2 and deﬁne for some λ > 0 21 A12 12 = A21 = 1, 12 A21 12 = A21 = −1, 11 22 22 A11 11 = A22 = A11 = A22 = λ, so that i j 2 Aαβ ij ξα ξβ = det(ξ) + λ|ξ| .
If u is merely continuous, we deﬁne its relaxed area, according to Lebesgue, as F(u) = inf lim inf F(uk ) uk → u uniformly, uk ∈ C 1 (Ω) . 27 Prove that the relaxed area functional is lower semicontinuous with respect to the uniform convergence. 28 The relaxed area functional agrees with the standard area functional on Lipschitz functions. 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) .
An Introduction to the Regularity Theory for Elliptic Systems, Harmonic Maps and Minimal Graphs by Mariano Giaquinta