Non-linear elliptic and parabolic equations involving measure data

Existence and uniqueness of “entropy” solutions of parabolic problems with L1 data

“…The method used, in [5], for showing existence of solutions for u 0 ∈ M (Ω) and f ∈ M ([0, T ] × Ω) consists in regularising u 0 and f with two sequences (u n 0 ) and…”

“…First, we explain how the solution of [5] is obtained and we precise its regularity and in the next two sections, we show existence and uniqueness of entropy solution.…”

Existence and uniqueness results for a nonlinear stationary system

“…With the aim of solving (1) with f (θ) ∈ L 2 (Ω) we are then led to assume that f satisfies the growth assumption ∀r ∈ R f (r) ≤ a + M |r| α , with a > 0, M > 0 and α < N/ 2(N − 2) if N ≤ 3 and α < ∞ if N = 2. Under this hypothesis on f , the coupling between Equations (1) and (2) together with the L. Boccardo and T. Gallouët estimates techniques (see [6] and Remark 5.5 of the present paper) lead to the following a priori estimate on θ,…”

“…u ∈ H 1 0 (Ω)) then the right-hand side of (2) belongs to L 1 (Ω). It follows from L. Boccardo and T. Gallouët [6] (see also [2] and [20]) that θ is expected in L q (Ω) for q < N/(N − 2) if N ≥ 3 and q < ∞ if N = 2. With the aim of solving (1) with f (θ) ∈ L 2 (Ω) we are then led to assume that f satisfies the growth assumption ∀r ∈ R f (r) ≤ a + M |r| α , with a > 0, M > 0 and α < N/ 2(N − 2) if N ≤ 3 and α < ∞ if N = 2.…”

Existence of entropy solutions for nonlinear elliptic degenerate anisotropic equations

“…As is generally known, an effective approach to the solvability of second-order equations in divergence form with L 1 -right-hand sides has been proposed in [1]. In this connection we also mention a series of other close investigations for nondegenerate isotropic nonlinear second-order equations with L 1 -data and measures, entropy and renormalized solutions [2][3][4][5][6][7][8][9][10].…”