Coercivity estimates for integro-differential operators
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Calculus of Variations
Coercivity estimates for integro-differential operators Jamil Chaker1 · Luis Silvestre1 Received: 7 May 2019 / Accepted: 30 April 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We provide a general condition on the kernel of an integro-differential operator so that its associated quadratic form satisfies a coercivity estimate with respect to the H s -seminorm. Mathematics Subject Classification 47G20 · 35R09 · 26A33
1 Introduction In this article, we are interested in coercivity estimates for integro-differential quadratic forms in terms of fractional Sobolev norms. More precisely, we seek general conditions on a kernel K (x, y) so that the following inequality holds for some constant c > 0 and any function u ∈ Hs, |u(x) − u(y)|2 K (x, y)dxdy ≥ cu2H˙ s . (1.1) Rd ×Rd
Here, H˙ s refers to the homogeneous fractional Sobolev norm whose standard expression is given by u2H˙ s = cd,s |u(x) − u(y)|2 |x − y|−d−2s dxdy = |u(ξ ˆ )|2 |ξ |2s dξ. Rd ×Rd
Rd
The quadratic form is naturally associated with the linear integro-differential operator (u(y) − u(x))K (x, y)dy. (1.2) Lu(x) = P V Rd
Communicated by O.Savin. Luis Silvestre is supported in part by NSF grant DMS-1764285. Jamil Chaker is supported by DFG Forschungsstipendium through Project 410407063.
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Jamil Chaker [email protected] Luis Silvestre [email protected]
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Mathematics Department, University of Chicago, Chicago, IL 60637, USA 0123456789().: V,-vol
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J. Chaker, L. Silvestre
Equations involving integro-differential diffusion like (1.2) have been the subject of intensive research in recent years. The understanding of the analog of the theorem of De Giorgi, Nash and Moser in the integro-differentiable setting plays a central role in the regularity of nonlinear integro-differential equations (See [7,8,11,13,14,17,18,20,21] and references therein). It concerns the generation of a Hölder continuity estimate for solutions of parabolic equations of the form u t = Lu, with potentially very irregular kernels K . There are diverse results in this direction with varying assumptions on K . The two key conditions that are necessary for this type of results are the coercivity condition (1.1) and the boundedness of the corresponding bilinear form: (u(y) − u(x))(v(y) − v(x))K (x, y)dxdy ≤ Cu H s v H s . (1.3) The initial works in the subject (like [8,21] or [11]) were focusing on kernels satisfying the convenient point-wise non-degeneracy assumption λ|x − y|−d−2s ≤ K (x, y) ≤ |x − y|−d−2s . These two inequalities easily imply (1.1) and (1.3). However, (1.1) and (1.3) hold under much more general assumptions. In [20] and [13], the coercivity estimate (1.1) is an assumption of the main theorem and some examples are given where the estimate applies to degenerate kernels. There are also recent applications of this framework to the Boltzmann equation (See [17]) where the kernels are not point-wise comparable to |x − y|−d−2s and yet (1.1) and (1.3) hold. While we know
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