By Mariano Giaquinta

ISBN-10: 8876424423

ISBN-13: 9788876424427

ISBN-10: 8876424431

ISBN-13: 9788876424434

This quantity bargains with the regularity concept for elliptic platforms. We could locate the starting place of this type of thought in of the issues posed via David Hilbert in his celebrated lecture introduced in the course of the overseas Congress of Mathematicians in 1900 in Paris: nineteenth challenge: Are the recommendations to standard difficulties within the Calculus of adaptations constantly inevitably analytic? twentieth challenge: does any variational challenge have an answer, only if sure assumptions concerning the given boundary stipulations are chuffed, and only if the idea of an answer is certainly prolonged? over the past century those difficulties have generated loads of paintings, often often called regularity concept, which makes this subject rather suitable in lots of fields and nonetheless very energetic for learn. in spite of the fact that, the aim of this quantity, addressed almost always to scholars, is way extra constrained. We goal to demonstrate just some of the elemental rules and strategies 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. themes contain: harmonic services, direct tools, Hilbert house equipment and Sobolev areas, power estimates, Schauder and L^p-theory either with and with out 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; 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 distinguished 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

40. The strong ellipticity condition gives the coercivity on W01,2 (Ω, Rm ) of the bilinear form B(v, w) := Ω i j Aαβ ij Dα v Dβ w dx, and the coercivity on W 1,2 (Ω, Rm ) of B(v, w) := Ω i j Aαβ ij Dα v Dβ w dx + γ v i ui dx. 31.

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).

Then by Jensen’s inequality |u(x)|p = p x1 Du(t, x)dt −a p a ≤ (2a)p Du(t, x)dt ≤ (2a)p−1 −a a −a |Du(t, x)|p dt. Integrating with respect to x ˜ and x1 yields Ω |u|p dx ≤ (2a)p−1 p−1 a dx = (2a) −a p −a Ω a dx1 |Du(t, x)|p dt Ω |Du|p dx p = (2a) Ω |Du| dx. The claim in the general case follows by density of Cc1 (Ω) in W01,p (Ω). 11 On the Banach space W01,p (Ω), Ω bounded and 1 ≤ p < ∞, the standard H 1,p norm is equivalent to u p 1,p W0 := Ω |Du|p dx. 12 There is a constant c = c(n, p) such that, if Ω ⊂ Rn is a convex set of diameter and u ∈ W 1,p (Ω), then Ω where uΩ = 1 |Ω| Ω u.

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An Introduction to the Regularity Theory for Elliptic Systems, Harmonic Maps and Minimal Graphs by Mariano Giaquinta


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