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Preprint, Working Paper, ... |
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| Subject: |
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Mathematics/Classical Analysis and ODEs
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| Title: |
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Algebra properties for Sobolev spaces- Applications to semilinear PDE's on manifolds |
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| Author(s): |
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Nadine Badr ( ) 1, Frederic Bernicot () 2, Emmanuel Russ () 3, 4 |
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| Laboratory: |
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| Abstract: |
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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. |
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| Fulltext language: |
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English |
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| Keyword(s): |
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Sobolev spaces – Riemannian manifold – algebra rule – paraproducts – heat semigroup |
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| Classification: |
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46E35 ; 22E30 ; 43A15 |
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| Comment: |
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29 pages |
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