By Mariano Giaquinta
This quantity offers with the regularity thought for elliptic structures. We might locate the foundation of this sort of thought in of the issues posed through David Hilbert in his celebrated lecture brought throughout the overseas Congress of Mathematicians in 1900 in Paris: nineteenth challenge: Are the options to commonplace difficulties within the Calculus of adaptations continually unavoidably analytic? twentieth challenge: does any variational challenge have an answer, only if definite assumptions in regards to the given boundary stipulations are chuffed, and only if the thought of an answer is certainly prolonged? over the past century those difficulties have generated loads of paintings, frequently often called regularity idea, which makes this subject relatively appropriate in lots of fields and nonetheless very energetic for examine. besides the fact that, the aim of this quantity, addressed normally to scholars, is far extra constrained. We target to demonstrate just some of the elemental rules and strategies brought during this context, confining ourselves to big yet easy events and refraining from completeness. in truth a few proper issues are passed over. issues contain: harmonic capabilities, direct equipment, Hilbert house tools and Sobolev areas, power estimates, Schauder and L^p-theory either with and with out strength 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 concept, together with the prestigious result of Wente and of Coifman-Lions-Meyer-Semmes.
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Extra info for An Introduction to the Regularity Theory for Elliptic Systems, Harmonic Maps and Minimal Graphs
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. In fact, ﬁx u ∈ H and L(v) := B(u, v). By Riesz theorem, L is uniquely represented by a vector which we call T u: B(u, v) = (T u, v). Observe that T is linear and continuous and deﬁne the symmetric, continuous and coercive bilinear form B(u, v) := (T ∗ u, T ∗ v).
Vn in Lp if for all i = 1, . . , n Ω uDi ϕdx = − Ω vi ϕdx ∀ϕ ∈ Cc∞ (Ω). It is easily seen that again weak derivatives are uniquely determined by u, if they exist, and that strong derivatives are also weak derivatives. 4 The class of functions u ∈ Lp (Ω) that possess weak derivatives in Lp is denoted by W 1,p (Ω). 5 Prove that H 1,p (Ω) ⊂ W 1,p (Ω). The following property is often used. 6 We say that an open set Ω ⊂ Rn has the extension property if for 1 ≤ p < ∞ and for any open set Ω Ω and every function u ∈ W 1,p (Ω) there exists u ∈ W 1,p (Ω) with u W 1,p (Ω) ≤ c(Ω, Ω) u W 1,p (Ω) .
52], . For instance, still assuming that 2 F the Hessian of F . Assume F = F (|p|), indicate with Fpα pβ := ∂p∂α ∂p β 2 that F is strictly convex and C , so that Fpi pj = Fpj pi and λ(p)|ξ|2 ≤ Fpα pβ (p)ξα ξβ ≤ Λ(p)|ξ|2 , for positive functions 0 < λ(p) ≤ Λ(p). Deﬁne the Bernstein function E(p) := Fpα pβ (p)pα pβ . 6) is possible. (i) lim sup |p|Λ(p) E(p) < ∞ |p|→∞ (ii) a. lim sup Λ(p) E(p) < ∞ and |p|→∞ b. the mean curvature of ∂Ω is non-negative. e. λ(p) ≥ γΛ(p) for every p ∈ Rn , and some γ > 0.
An Introduction to the Regularity Theory for Elliptic Systems, Harmonic Maps and Minimal Graphs by Mariano Giaquinta