Page 131 - Textos de Matemática Vol. 34

THE GEOMETRY OF SOME NONLINEAR PDES JOSE´ MIGUEL URBANO
Dedicated to Professor J. A. Sampaio Martins on the occasion of his sixtieth birthday
1. Introduction
Considerable progress was made in the early 1950s and mid 1960s in the theory of elliptic equations, due to the discoveries of DeGiorgi [4] and Moser [15], [16]. Consider local weak solutions of
(1.1) a u =0 weaklyinΩ; u∈H1 (Ω), ij xj xi loc
where Ω ⊂ RN and the coefficients x → aij(x), i,j = 1,2,...,N are only bounded and measurable and satisfy an ellipticity condition. DeGiorgi established that lo- cal solutions are H¨older continuous and Moser proved that non-negative solutions satisfy the Harnack inequality. Such inequality can be used, in turn, to prove the H¨older continuity of solutions. Both authors worked with linear PDEs. However the linearity has no bearing in the proofs. This permits an extension of these results to elliptic quasilinear equations of the type
(1.2) divA(x,u,∇u)+B(x,u,∇u)=0 weaklyinΩ; u∈W1,p(Ω); p>1 loc
with structure conditions
C0|∇u|p−C≤A(x,u,∇u)·∇u;
(1.3) |A(x, u, ∇u)| + |B(x, u, ∇u)| ≤ C1 + |∇u|p−1 ,
for a given C0 > 0 and a given non–negative constant C. By using the methods of DeGiorgi, Ladyzhenskaja and Ural’ceva [13] established that weak solutions of (1.2)–(1.3) are H¨older continuous, whereas Serrin [19] and Trudinger [20], following the methods of Moser, proved that non-negative solutions satisfy a Harnack prin- ciple. The generalisation is twofold: the principal part A(x, u, ∇u) is permitted to have a nonlinear dependence with respect to ∇u, and a nonlinear growth with
2000 Mathematics Subject Classification. 35B65, 35D10, 35K65.
Key words and phrases. Degenerate and singular PDEs, intrinsic scaling, (Ho¨lder) continuity. Research supported by CMUC/FCT and Project POCTI/34471/MAT/2000.
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