Algebra properties for Sobolev spaces- Applications to semilinear PDE's on manifolds

Nadine Badr (1) , Frederic Bernicot (2) , Emmanuel Russ (3, 4)

1 ICJ - Institut Camille Jordan
2 LPP - Laboratoire Paul Painlevé - UMR 8524
3 LATP - Laboratoire d'Analyse, Topologie, Probabilités
4 IF - Institut Fourier

In this work, we aim to prove algebra properties for generalized Sobolev spaces $W^{s,p} \cap L^\infty$ on a Riemannian manifold, where $W^{s,p}$ is of Bessel-type $W^{s,p}:=(1+L)^{-s/m}(L^p)$ with an operator $L$ generating a heat semigroup satisfying off-diagonal decays. We don't require any assumption on the gradient of the semigroup. To do that, we propose two different approaches (one by a new kind of paraproducts and another one using functionals). We also give a chain rule and study the action of nonlinearities on these spaces and give applications to semi-linear PDEs. These results are new on Riemannian manifolds (with a non bounded geometry) and even in the Euclidean space for Sobolev spaces associated to second order uniformly elliptic operators in divergence form.

Mots clés: Sobolev spaces; Riemannian manifold; algebra rule; paraproducts; heat semigroup

Domaines: Analyse classique [math.CA]

Nadine Badr, Frederic Bernicot, Emmanuel Russ. Algebra properties for Sobolev spaces- Applications to semilinear PDE's on manifolds. Journal d'analyse mathématique, 2012, 118 (2), pp.509-544. ⟨hal-00609697⟩