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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.

We establish a new oscillation estimate for solutions of nonlinear partial differential equations of elliptic, degenerate type. This new tool yields a precise control on the growth rate of solutions near their set of critical points, where ellipticity degenerates. As a consequence, we are able to prove the planar counterpart of the longstanding conjecture that solutions of the degenerate p-Poisson equation with a bounded source are locally of class C p ′ = C 1, 1 p−1 ; this regularity is optimal.

We give a proof of the Hölder continuity of weak solutions of certain degenerate doubly nonlinear parabolic equations in measure spaces. We only assume the measure to be a doubling non-trivial Borel measure which supports a Poincaré inequality. The proof discriminates between large scales, for which a Harnack inequality is used, and small scales, that require intrinsic scaling methods. 2000 Mathematics Subject Classification. Primary 35B65. Secondary 35K65, 35D10. Key words and phrases. Hölder continuity, Caccioppoli estimates, intrinsic scaling, Harnack's inequality. Research of JMU supported by CMUC/FCT and project UTAustin/MAT/0035/2008. 1 2 KUUSI, SILJANDER AND URBANO two complementary cases: Case I : 0 ≤ ess inf u << ess osc u and Case II : u p−2 u t ≈ Cu t .In large scales, i.e., in Case I, the scaling property of the equation dominates and, consequently, the reduction of the oscillation follows immediately from Harnack's inequality. In small scales, on the other hand, the oscillation is already very small and thus the solution itself is between two constants, the infimum and the supremum, whose difference is negligible. Correspondingly, the nonlinear time derivative term, which formally looks like u p−2 u t , behaves like u t and we end up with a p-parabolic type behavior. However, also in this case, we still need to modify the known arguments. In particular, the energy estimates are not available in the usual form and we need to use modified versions as in [5], [15] and [25].Our argument also applies to doubly nonlinear equations of p-Laplacian type that are of the formwith A(x, t, ·, ·) satisfying the usual structure assumptions. For expository purposes, we only consider (1.1). Very recently, a direct geometric method to obtain local Hölder continuity for parabolic equations has been developed in [6] and [9]. Despite the effort, the general picture remains unclear. Preliminaries

To Emmanuele DiBenedetto, with admiration and friendship, on the occasion of his 70th birthday.Abstract. We provide a broad overview on qualitative versus quantitative regularity estimates in the theory of degenerate parabolic pdes. The former relates to DiBenedetto's revolutionary method of intrinsic scaling, while the latter is achieved by means of what has been termed geometric tangential analysis. We discuss, in particular, sharp estimates for the parabolic p−Poisson equation, for the porous medium equation and for the doubly nonlinear equation.

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We prove existence and uniqueness results for weak solutions of nonlinear degenerate problems arising in various physical models. The main novelty in the article concerns the uniqueness, which employs a technique based in showing that weak solutions are also entropy solutions, for which uniqueness follows from a straightforward adaptation of known results. We treat equations with lower order terms that have a particular structure and show with a counterexample that for general lower order terms the uniqueness does not hold.2000 Mathematics Subject Classification: 35D99, 35K65, 35R35.

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