Page 133 - Textos de Matemática Vol. 34

THE GEOMETRY OF SOME NONLINEAR PDES 123
2. The classical approach of DeGiorgi
Grasping the meaning of intrinsic scaling calls for a full understanding of the basic ideas behind the approach of DeGiorgi to the regularity of the solutions of linear equations with coefficients that are merely measurable and bounded. The two main ingredients are energy estimates on level sets and an iteration argument.
Let Ω ⊂ RN be a smooth bounded domain and consider the equation (2.1) a u =0 in Ω,
ij xj xi
where the coefficients aij(x) are only assumed to be measurable and bounded, with
∥aij ∥L∞ ≤ Λ, and to satisfy the uniform ellipticity condition aij(x)ξiξj ≥λ|ξ|2 , ∀x∈Ω, ∀ξ∈RN ,
for some λ > 0. A weak solution of equation (2.1) is a function u ∈ H1(Ω) which satisfies the integral identity
∂u∂φ
(2.2) aij∂x∂x =0,
Ωij for all test functions φ ∈ H01(Ω).
Theorem 2.1. Every weak solution of (2.1) is locally bounded. Proof. To simplify the writing assume that
Ω=B1 :=x∈RN : |x|<1 .
Let k ≥ 0 and η be a smooth function with compact support in B1. Put v = (u−k)+ and take φ = vη2 as test function in (2.2). The use of the assumptions and Young’s inequality give
22 4Λ2 22 (2.3) |∇v|η ≤ λ2 |∇η|v .
B1 B1
These energy inequalities on level sets of u will be the building blocks of the whole theory and once they are obtained the PDE can be forgotten: the problem becomes purely analytic.
Next, by H¨older and Sobolev’s inequalities (denoting with 2∗ = 2N/(N − 2) the Sobolev exponent),
2
(vη)2 ≤ B1
≤
∗2∗ 2 (vη)2 |{vη ̸= 0}|1− 2∗
B1
c(N) |{vη ̸= 0}|N
22 |∇(vη)|
B1