Sequences of Laplacian Cut-Off Functions

Güneysu, Batu

doi:10.1007/s12220-014-9543-9

Cited by 36 publications

(59 citation statements)

References 23 publications

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“…Proposition 2.4 (Theorem 2.6 in [17] and Proposition 3.6 in [20]). (a) Assume that (CZ(p)) holds for some 1 < p < ∞ and that M admits a sequence of Laplacian cut-off functions.…”

Section: Sequences Of Cut-off Functions and Applications To Density Pmentioning

confidence: 97%

“…Sequences of Laplacian and Hessian cut-off functions where defined in [17] and [20]. Here we will need to introduce also the slightly different notions of weak Laplacian and weak Hessian cut-off functions.…”

Section: Sequences Of Cut-off Functions and Applications To Density Pmentioning

confidence: 99%

“…The following proposition (partially taken from [18]) should give the state of the art on the existence of such sequences of cut-off functions: point (i) follows by [31] (see also [17] for an alternative proof in the case C = 0), point (ii) was proved in [8] .…”

Section: Sequences Of Cut-off Functions and Applications To Density Pmentioning

confidence: 99%

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Density Problems for second order Sobolev Spaces and Cut-off Functions on Manifolds With Unbounded Geometry

Impera

Rimoldi

Veronelli

2019

International Mathematics Research Notices

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We consider complete non-compact manifolds with either a sub-quadratic growth of the norm of the Riemann curvature, or a sub-quadratic growth of both the norm of the Ricci curvature and the squared inverse of the injectivity radius. We show the existence on such a manifold of a distance-like function with bounded gradient and mild growth of the Hessian. As a main application, we prove that smooth compactly supported functions are dense in W 2,p . The result is improved for p = 2 avoiding both the upper bound on the Ricci tensor, and the injectivity radius assumption. As further applications we prove new disturbed Sobolev and Calderón-Zygmund inequalities on manifolds with possibly unbounded curvature and highlight consequences about the validity of the full Omori-Yau maximum principle for the Hessian. ContentsProblem 1.1. Under which (geometric) assumptions on M does one have that W 2,p 0 (M ) = W 2,p (M )? Classical results on this topic can be found in [6], [24] and references therein. In the following proposition we collect the most up-to-date achievements: point (I) was shown by E. Hebey, [23, Theorem 2.8]; point (II) was proved by B. Güneysu in [18, Proposition III.18]; point (III) is due to L. Bandara, [7] (for an alternative proof see also [18, Proposition III.18]). Proposition 1.2. Let (M m , g) be a complete Riemannian manifold.

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“…Proposition 2.4 (Theorem 2.6 in [17] and Proposition 3.6 in [20]). (a) Assume that (CZ(p)) holds for some 1 < p < ∞ and that M admits a sequence of Laplacian cut-off functions.…”

Section: Sequences Of Cut-off Functions and Applications To Density Pmentioning

confidence: 97%

Section: Sequences Of Cut-off Functions and Applications To Density Pmentioning

confidence: 99%

See 1 more Smart Citation

Density Problems for second order Sobolev Spaces and Cut-off Functions on Manifolds With Unbounded Geometry

Impera

Rimoldi

Veronelli

2019

International Mathematics Research Notices

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“…On the other hand, it was proved in [19,Theorem 2.2] that one can construct families of cutoff functions {φ R } with a Euclidean like behavior of |∇φ R | and |∆φ R | in terms of R, provided the Ricci curvature is nonnegative. This paper started trying to extend the results obtained in [6], by M.Bonforte, G. Grillo and L. Vazquez, where they consider Cartan-Hadamard manifolds with Ricci curvature (and therefore sectional curvture) bounded from below, under relaxed geometric assumption.…”

mentioning

confidence: 99%

“…The last two sections are devoted to applications. In Section 3 we present a first direct application of the existence of sequences of Laplacian cut-offs to obtain a generalization of the L q -properties of the gradient and the self-adjointness of Schrödinge-type operators discussed in [32] and [19] to to the class of Riemannian manifolds satisfying our more general Ricci curvature conditions. Section 4 is arguably the second main part of the paper.…”

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confidence: 99%

Laplacian cut-offs, porous and fast diffusion on manifolds and other applications

Bianchi

Setti

2017

Calc. Var.

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We construct exhaustion and cut-off functions with controlled gradient and Laplacian on manifolds with Ricci curvature bounded from below by a (possibly unbounded) nonpositive function of the distance from a fixed reference point, without any assumptions on the topology or the injectivity radius. Along the way we prove a generalization of the Li-Yau gradient estimate which is of independent interest. We then apply our cut-offs to the study of the fast and porous media diffusion, of L q -properties of the gradient and of the self-adjointness of Schrödinger-type operators. Introduction.Many analytic results in Euclidean setting require the use of compactly supported cut-off functions, essentially to localize differential equations or inequalities or to perform integration by parts arguments. A key feature of d-dimensional Euclidean space is that it is possible to construct cut-offs {φ R } such that φ R = 1 on the ball B R (o), they are supported in the ball B γR (o) *

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Self‐adjointness of perturbed biharmonic operators on Riemannian manifolds

Milatovic

2017

Mathematische Nachrichten

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We give a sufficient condition for the essential self-adjointness of a perturbed biharmonic-type operator acting on sections of a Hermitian vector bundle on a geodesically complete Riemannian manifold with Ricci curvature bounded from below by a (possibly unbounded) non-positive function depending on the distance from a reference point. We also establish the separation property in the case when the corresponding operator acts on functions. K E Y W O R D SBiharmonic operator, Bochner Laplacian, perturbation, Riemannian manifold, self-adjointness M S C ( 2 0 1 0 ) 35P05, 47B25, 58J05, 58J50 INTRODUCTIONThe question of self-adjointness of differential operators in 2 (ℝ ) has been in the focus of attention of researchers for a long time. In particular, Schrödinger operators have occupied a special place because of their importance in mathematical physics. Over many decades now, various sufficient conditions for the (essential) self-adjointess of Schrödinger operators have been established; see the books [12,22,31] for a discussion of related results. Parallel to numerous developments in the realm of Schrödinger operators, past few decades have witnessed quite a bit of activity concerning the self-adjointness of higher order operators in 2 (ℝ ); see, for instance, [9,23,24,27] and references therein.The study of self-adjointness of differential operators on Riemannian manifolds was initiated in the landmark paper [14]. Subsequently, the authors of [10] and [11] established the self-adjointness of positive integer powers of the scalar Laplacian (and Laplacian on differential forms). In the last two decades, there has been a lot of activity on the problem of self-adjointness of Schrödinger operators on Riemannian manifolds (including operators acting on sections of Hermitian vector bundles), as seen, for instance, in [2,[6][7][8]16,17,20,25,29,30,32]. Recently, the authors of [3] proved, among other things, the essential selfadjointness of powers of first-order elliptic operators with low-regularity coefficients on vector bundles with low-regularity coefficient metrics over manifolds with low-regularity metrics.Our paper is concerned with the (essential) self-adjointness of ( ∇ † ∇ ) 2 + , the square of the Bochner Laplacian plus a potential ; see Section 2.1 for an explanation of notations. Methodologically, our problem is handled similarly as in [27], with modifications in some estimates due to the geometry of the manifold and nature of our operator. To help us with our estimates, we use a family of cut-off functions constructed recently by the authors of [4] in the context of a geodesically complete Riemannian manifold with Ricci curvature bounded from below by a certain non-positive function; see the assumption (R) in Section 2 below

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Sequences of Laplacian Cut-Off Functions

Cited by 36 publications

References 23 publications

Density Problems for second order Sobolev Spaces and Cut-off Functions on Manifolds With Unbounded Geometry

Density Problems for second order Sobolev Spaces and Cut-off Functions on Manifolds With Unbounded Geometry

Laplacian cut-offs, porous and fast diffusion on manifolds and other applications

Self‐adjointness of perturbed biharmonic operators on Riemannian manifolds

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