Quasi-linear degenerate elliptic problems with data

Nonlinear elliptic boundary value problem with equivalued surface in a domain with thin layer

“…Hence we are not able to prove the existence and uniqueness of weak solutions in the sense of Definition 2.1 to problem (P 1 ). In order to overcome this difficulty, we will use the concept of renormalized solution introduced by Diperna and Lions in [3] for Boltzmann equations (see also [1,2,7]). The renormalized solution is different from the weak solution or the distribution solution because it is defined by truncation functions.…”

“…(2) Proof of uniqueness Here the idea of this proof comes from [1]. Let u 1 , u 2 ∈ V 1 be two renormalized solutions to problem (P 1 ) in the sense of Definition 2.3.…”

Uniqueness of solutions for some elliptic equations with a quadratic gradient term

“…Other uniqueness results for equations with a gradient term can be found in [31], where the assumption β(s) ≤ 0 is considered, [26] for uniqueness of the zero solution for a sign-changing nonsingular function β and f (x, s)s ≤ 0, [8] for the case of subquadratic terms in ∇u, [11][12][13] for equations with different dependence on ∇u and [14,22,29] for quasilinear equations with no quadratic term in ∇u.…”

Nonlinear equations with unbounded heat conduction and integrable data

“…Conditions (1.10) and (1.13) are classical when dealing with renormalized solutions for partial differential equations with L 1 data (see [12,13,2]). The fact that u ≤ m almost everywhere in Ω is already explained (and is natural) in [6].…”

“…(1) Laboratoire Jacques-Louis Lions, Université Paris VI, Boîte courrier 187, 75252 Paris cedex 05 (2) We investigate a class of diffusion problems, in the stationary and evolution cases, with singular matrices with respect to the unknown. More precisely let Ω be a bounded domain of R N , m and T be two positive real numbers and β and γ be two functions of C 0 ((−∞, m)) such that β(s) ≤ γ(s), lim s→m − β(s) = +∞ and m 0 γ(s) ds < +∞.…”