Quasi-Linear Relaxed Dirichlet Problems
Stefano Finzi Vita, François Murat, Nicoletta Anna Tchou · SIAM Journal on Mathematical Analysis · 1996
We study the existence and the asymptotic behavior of solutions of quasi-linear elliptic problems with homogeneous Dirichlet boundary conditions for the so-called relaxed Dirichlet problems. These problems, introduced in the linear case by Dal Maso and Mosco [Arch. Rational Mech. Anal., 95 (1986), pp. 345–387; Appl. Math. Optim., 15 (1987), pp. 15–63], involve zero-order terms with Borel measures, which can take infinite values but vanish on sets with zero capacity. We prove two existence results when the nonlinear term has quadratic growth with respect to the gradient by extending the techniques of Boccardo, Murat, and Puel [Res. Notes in Math. 84, Pitman, London, 1983, pp. 19–73] to the relaxed case. We also prove in the subquadratic case a stability property for bounded solutions with respect to the $\gamma $-convergence of measures when the limit measure is sufficiently regular, making essential use of the correctors result of Finzi Vita and Tchou [Asymptotic Anal., 5 (1992), pp. 269–281].