Entropy solutions for the $p(x)$-Laplace equation

Abstract

Abstract. We consider a Dirichlet problem in divergence form with variable growth, modeled on the p(x)-Laplace equation. We obtain existence and uniqueness of an entropy solution for L 1 data, as well as integrability results for the solution and its gradient. The proofs rely crucially on a priori estimates in Marcinkiewicz spaces with variable exponent, for which we obtain new inclusion results of independent interest.

Gradient estimates via non standard potentials and continuity

“…Note that there is in general no uniqueness in W 1,q(·) ; see the counterexamples already found for the constant coefficient case in [50,46]. Concerning the existence of a SOLA we note that, since we are assuming p(·) ≥ 2, in particular cases (for instance when n ≤ 10, or p(·) ≡ const) the entropy solution constructed in [49,Theorem 1.3] or [8, Theorem 2.1] coincides with the SOLA (see [49,Remark 5.7]), and hence we can infer existence from there. However, in our case of exponents p(·) ≥ 2 lying above the critical exponent 2 − 1/n the existence theory is essentially easier and we can infer existence-and uniqueness when µ ∈ L…”

“…(Ω) the general existence theory of weak solutions does not apply and therefore one has to consider a more general notion of solution. In the literature there are different approaches, see [8,40,49]. We shall follow the one introduced in [9,10,14] (Ω) and therefore the weak formulation of (1.11) makes sense for a SOLA.…”

“…(Ω)-of a SOLA in the case of coefficients with non standard growth in the spirit of [9,10,14] via the methods of [8,40,49]. For the sake of briefness we shall not give the whole proof here, but only the main steps and the approximation procedure leading to the gradient estimate (1.10) for a SOLA.…”

Existence of entropy solutions for some nonlinear elliptic problems involving variable exponent and measure data

“…Using the same argument in ( [22], Proposition 5.1), we can prove that (u n ) n is a Cauchy sequence in measure, then there exists a subsequence still denoted (u n ) n such that u n → u a.e. in Ω, we conclude that…”

Nonlinear diffusion equations driven by the p(·)-Laplacian

“…Equation (1.1) was also studied by Bendahmane et al [15], where the well-posedness is proved for renormalized solutions in an L 1 -framework for continuous variable exponents. Moreover, the existence and uniqueness of entropy solutions and the equivalence between two notions of solutions are discussed for log-Hölder continuous variable exponents in [38] by Zhang and Zhou (see also [36] for an elliptic counterpart).…”