کد مقاله | کد نشریه | سال انتشار | مقاله انگلیسی | نسخه تمام متن |
---|---|---|---|---|
6417947 | 1339319 | 2015 | 31 صفحه PDF | دانلود رایگان |

We show how Lasry-Lions's result on regularization of functions defined on Rn or on Hilbert spaces by sup-inf convolutions with squares of distances can be extended to (finite or infinite dimensional) Riemannian manifolds M of bounded sectional curvature. More specifically, among other things we show that if the sectional curvature K of M satisfies âK0â¤Kâ¤K0 on M for some K0>0, and if the injectivity and convexity radii of M are strictly positive, then every bounded, uniformly continuous function f:MâR can be uniformly approximated by globally C1,1 functions defined by(fλ)μ=supzâMâ¡infyâMâ¡{f(y)+12λd(z,y)2â12μd(x,z)2} as λ,μâ0+, with 0<μ<λ/2. Our definition of (global) C1,1 smoothness is intrinsic and natural, and it reduces to the usual one in flat spaces, but we warn the reader that, in the noncompact case, this definition differs from other notions of (rather local) C1,1 smoothness that have been recently used by other authors (based on charts). The importance of this regularization method lies (rather than on the degree of smoothness obtained) on the fact that the correspondence fâ¦(fλ)μ is explicit and preserves many significant geometrical properties that the given functions f may have, such as invariance by a set of isometries, infima, sets of minimizers, ordering, local or global Lipschitzness, and (only when one additionally assumes that Kâ¤0) local or global convexity. We also give two examples showing that this result completely fails, even for (nonflat) Cartan-Hadamard manifolds, whenever f or K are not bounded.
Journal: Journal of Mathematical Analysis and Applications - Volume 423, Issue 2, 15 March 2015, Pages 994-1024