Uniform estimates for some degenerating quasilinear elliptic equations and a bound on the Harnack constant for linear equations

Article type: Research Article

Authors: Bardi, Martino^; | Perthame, Benoît

Affiliations: Dipartimento di Matematica Pura e Applicata, Universita di Padova, Via Belzoni 7, I-35131 Padova, Italy | Département de Mathématiques, Université d'Orléans, BP 6759, F 45067 Orléans Cedex 2, France

Note: [] This work was done while this author was visiting CEREMADE, Université Paris-Dauphine, partially supported by the Consiglio Nazionale delle Ricerche.

Abstract: Let u be any smooth solution of −εaijuxixj+aijuxiuxj+biuxi=εc in B1={x:|x|<1}, where ε is a small parameter. We prove some uniform estimates on u in W1,q under weak assumptions on the coefficients aij, bi and c. This is motivated by the study of exponential behaviors for solutions of linear elliptic equations −εaijvxixj+bivxi+cv=0, as ε goes to 0. For this equation, our result provides an estimate of Harnack type sup Brv≤Cinf Brv, with C=e−krα/ε for some α≤1.

DOI: 10.3233/ASY-1991-4101

Journal: Asymptotic Analysis, vol. 4, no. 1, pp. 1-16, 1991