By Mariano Giaquinta
This quantity bargains with the regularity concept for elliptic platforms. We may possibly locate the beginning of this sort of concept in of the issues posed by way of David Hilbert in his celebrated lecture introduced throughout the overseas Congress of Mathematicians in 1900 in Paris: nineteenth challenge: Are the suggestions to typical difficulties within the Calculus of diversifications regularly inevitably analytic? twentieth challenge: does any variational challenge have an answer, only if convinced assumptions concerning the given boundary stipulations are chuffed, and only if the inspiration of an answer is definitely prolonged? over the last century those difficulties have generated loads of paintings, often known as regularity conception, which makes this subject relatively appropriate in lots of fields and nonetheless very energetic for examine. even though, the aim of this quantity, addressed often to scholars, is far extra restricted. We target to demonstrate just some of the fundamental rules and methods brought during this context, confining ourselves to special yet basic events and refraining from completeness. actually a few suitable issues are passed over. issues contain: harmonic features, direct tools, Hilbert house tools and Sobolev areas, power estimates, Schauder and L^p-theory either with and with out capability 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; 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 concept, together with the distinguished 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
E. ] Bilinear symmetric forms Suppose B is a symmetric, continuous and coercive bilinear form on H, where continuous and coercive respectively mean that there exist Λ, λ > 0 such that |B(u, v)| ≤ Λ u v , B(u, u) ≥ λ u 2 , for all u, v ∈ H. Then B is a scalar product equivalent to the original (·, ·) and the Dirichlet principle applies, giving the following theorem. 2 The functional F(u) = 1 B(u, u) − L(u) 2 has a unique minimizer u. Moreover u satisﬁes B(u, v) = L(v) for each v ∈ H. 2 without the symmetry assumption is known as Lax-Milgram’s theorem.
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.
Then u ≤ v in Ω. Proof. It is enough to prove the claim for w = v + ε instead of v, and let ε → 0. By the comparison principle it suﬃces to show that u≤w on ∂ 0 Ω. If not, there exists t > 0 as small as we want such that γt := sup (u − w) > 0 A∩Γt and u − w ≤ 0 on Γt \A. 9) there exists x0 ∈ Γt ∩ A with u(x0 ) = w(x0 ) + γt and thus ∂ ∂ (u − v)(x0 ) = (u − w)(x0 ) ≥ 0. ∂ν ∂ν Since t > 0 can be chose arbitrarily small, we found a contradiction to hypothesis 2. 22 For every ε > 0 there exists a neighborhood Γ of x0 in ∂Ω such that if u minimizes F in the class A, then ε sup |u| ≤ sup |g| + .
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