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We study well-posedness for elliptic problems under the form b(u) − div a(x, u,∇u) = f, where a satisfies the classical Leray-Lions assumptions with an exponent p that may depend both on the space variable x and on the unknown solution u. A prototype case is the equation u − div | ∇u| p(u)−2 ∇u = f. We have to assume that inf x∈Ω, z∈R p(x, z) is greater… (More)
In this article, we prove the existence and uniqueness of solutions for a family of discrete boundary value problems for data f which belongs to a discrete Hilbert space W.
We study the Cauchy problem associated with the nonlinear el-liptic-parabolic equation b(u)t − a(u, ϕ(u)x)x = f. We prove an L 1-contraction principle and hence the uniqueness of entropy solutions, under rather general assumptions on the data.
We study the structural stability (i.e., the continuous dependence on coefficients) of solutions of the elliptic problems under the form b(u n) − div a n (x,∇u n) = f n. The equation is set in a bounded domain Ω of R N and supplied with the homogeneous Dirichlet boundary condition on ∂Ω. Here b is a non-decreasing function on R, and a n (x, ξ) n is a family… (More)
In this article we study the nonlinear homogeneous Neumann boundary-value problem b(u) − div a(x, ∇u) = f in Ω a(x, ∇u).η = 0 on ∂Ω, where Ω is a smooth bounded open domain in R N , N ≥ 3 and η the outer unit normal vector on ∂Ω. We prove the existence and uniqueness of a weak solution for f ∈ L ∞ (Ω) and the existence and uniqueness of an entropy solution… (More)