Nonlinear problems with unbounded coefficients and $$L^1$$ L 1 data
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Nonlinear Differential Equations and Applications NoDEA
Nonlinear problems with unbounded coefficients and L1 data Filomena Feo
and Olivier Guib´e
Abstract. We consider a class of nonlinear elliptic problems whose prototype involves a coefficients matrix blowing up for a finite value m of the unknown u. Since datum is in L1 , a suitable notion of renormalized solutions is introduced taking into account that u can reach the value m and the existence of such solutions is proved. We study the corresponding evolution problem as well. Mathematics Subject Classification. 35J66, 35K61. Keywords. Nonlinear equations, Blow-up, Existence, Renormalized solutions, Integrable data.
Introduction In this paper we deal with a class of nonlinear problems whose prototype for 1 < p < N is p−2 − div [(A(x, u) Du · Du) 2 A(x, u) Du] = f in Ω, (0.1) u = 0 on ∂Ω. Here Ω is a bounded domain of RN , f ∈ L1 (Ω) and A is function that associates a symmetric matrix A(x, u) to pair (x, u) that blows up (uniformly in ×N x) for finite value of the unknown. Precisely m > 0, A : Ω×(−∞, m) → RN S Carath´eodory function such that there exist β, γ ∈ C 0 ((−∞, m), (0, +∞)) verifying γ(s) ≥ β(s) ≥ α > 0, ∀s ∈ (−∞, m), lims→m− β(s) = +∞ and 2
2
β p (s)|ξ|2 ≤ A(x, s)ξ · ξ ≤ γ p (s)|ξ|2 a.e. in Ω N
(0.2)
∀s ∈ (−∞, m), ∀ξ ∈ R . In literature similar problems have been studied under different assumptions. We refer the readers for example to [2,7–10,16] and [17]. The main difficulties are due to the blow up of the matrix, to the possible different behaviour of β and γ as s → m− and to the L1 −datum. In order to 0123456789().: V,-vol
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F. Feo and O. Guib´ e
NoDEA
detail them, let us consider first for simplicity the datum in the dual space and β equals to γ up a constant. Even in this case we have to give sense of field A(x, u) Du on the set {x ∈ Ω : u(x) = m} (see [5] for an extensive treatment of this topic in the case of diagonal matrix and p = 2). It is well-known that we have different possible behaviours depending on the summability of γ. If 1 γ p−1 ∈ L1 (0, m), then the set {x ∈ Ω : u(x) = m} has zero Lebesgue measure 1 and there exists a weak solution. Otherwise when γ p−1 ∈ L1 (0, m), the set {x ∈ Ω : u(x) = m} can have a positive Lebesgue measure and there is no weak solution if we don’t require the smallness of the Lebesgue norm of datum. Moreover as in our prototype problem (0.1) if the lower bound and the upper bound in (0.2) are expressed in term of two positive function blow ways up in two possible different way as s → m− , we can not obtain the usual u 1 u 1 a priori estimates. Indeed if formally we take 0 β p−1 (t) dt and 0 γ p−1 (t) dt 1 u as test functions we obtain an Lp - estimate of gradient of 0 β p−1 (t) dt by p the lower bound of (0.2), but in general not an L estimate of (A(x, u) Du · u 1 p−2 Du) 2 A(x, u) Du (nor 0 γ p−1 (t) dt) by the upper bound of (0.2) as happens when β(s) = γ(s) up to a constant. Then all detailed difficulties and the presence of L1 datum constrain us to give a suitable definition of renormalized
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