Existence and uniqueness of entropy solutions for nonlinear elliptic equations with measure data

Abstract

L'accès aux archives de la revue « Annales de l'I. H. P., section C » (http://www.elsevier.com/locate/anihpc) implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/

Nonlinear Muckenhoupt–Wheeden type bounds on Reifenberg flat domains, with applications to quasilinear Riccati type equations

“…We denote by M 0 (Ω) (respectively M s (Ω)) the set of all measures in M B (Ω) which are absolutely continuous (respectively singular) with respect to the capacity cap p (·, Ω). Here cap p (·, Ω) is the capacity relative to the domain Ω defined by The notion of renormalized solutions is a generalization of that of entropy solutions introduced in [2] and [4], where the measure data are assumed to be in L 1 (Ω) or in M 0 (Ω). Several equivalent definitions of renormalized solutions were given in [8], two of which are the following ones.…”

“…The uniqueness is guaranteed here since |∇v| q + ω ∈ M 0 (Ω) (see [4,8]). Also, note that S(v) ∈ E for every v ∈ E since by Theorem 5.6 and the fact that T is a root of g we have By Lemma 5.8 below the map S : E → E is continuous and S(E) is precompact under the strong topology of W 1, 1 0 (Ω).…”

Semilinear elliptic equations and systems with diffuse measures

“…According to a result of BoccardoGallouët-Orsina [2], µ is diffuse if, and only if, µ ∈ L 1 (Ω) + H −1 (Ω), i.e.…”

Local Properties of Solutions of Elliptic Equations Depending on Local Properties of the Data

“…In [1] (see also [4]), a notion of solution for (1.2) has been introduced if f ∈ L 1 (Ω) and the function a(x, s, ξ) does not depend on s, so that the differential operator A is strictly monotone, with the purpose of proving its uniqueness: the so-called entropy solution. In this case, the strong limit u in W 1,q 0 (Ω), q < N N −1 , of the sequence {u n } is the unique entropy solution of (1.2), so that Theorem 1.1 can be seen as giving improved summability properties of the entropy solution.…”