Lq–L∞ Hölder continuity for quasilinear parabolic equations associated to Sobolev derivations

Let X be a locally compact metrizable space endowed with a couple of equivalent finite Radon measures m and μ and let E be a Hilbert C∗-monomodule over C0(X). We consider a class of abstract nonlinear parabolic equations defined as follows. Let ∂ be a closed derivation from L2(X,m) to L2(E,μ) and Tt be the strongly continuous nonlinear semigroup naturally associated, in the sense of Brezis (1973), to the convex l.s.c. functional E(u) = X |∂u|px dμ(x), where | · | is the natural modulus function associated to E. The generator of the semigroup considered is a natural generalization of the usual p-Laplacian operator. We suppose that a suitable Sobolev-like inequality of the form u L2d/(d−2)(X,m) c ∂u L2(E,μ) holds true for some d > 2, with p ∈ [2,d). Then Tt is a nonlinear Markov semigroup in the sense that it is order preserving and nonexpansive on each Lq(X,m) for any q ∈ [2,+∞] and, moreover, it satisfies Tt u − Tt v L∞(X,m) cm(X)αt−β u − v γ Lq(X,m) for all q 2 and suitable constants α, β, γ depending only on p, q, d. Examples include the semigroup generated by the p-Laplacian on finite measure manifolds with boundary and homogeneous Dirichlet boundary conditions, as well as by p-Laplacian-like operators associated both to regular sub-Riemannian structures, andto systems of (possibly singular or degenerate) vector fields satisfying the appropriate Sobolev inequalities.  2002 Elsevier Science (USA). All rights reserved.