## Sharp Estimates for Iterated Green Functions (2002)

Venue: | PROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH A 132 (2002), 91–120. |

Citations: | 10 - 1 self |

### BibTeX

@MISC{Grunau02sharpestimates,

author = {H.-ch. Grunau and G. Sweers},

title = {Sharp Estimates for Iterated Green Functions },

year = {2002}

}

### OpenURL

### Abstract

Optimal pointwise estimates from above and below are obtained for iterated (poly)harmonic Green functions corresponding to zero Dirichlet boundary conditions. For second order elliptic operators these estimates hold true on bounded C 1,1 domains. For higher order elliptic operators we have to restrict ourselves to the polyharmonic operator on balls. We will also consider applications to noncooperatively coupled elliptic systems and to the lifetime of conditioned Brownian motion.

### Citations

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Citation Context ...ucial tosnd optimal pointwise estimates for the corresponding Green functions. And indeed, in recent years such estimates have been developed. Motivated by Schrodinger operators Zhao ([21], see also [=-=4]-=-) was in 1986 thesrst to prove a sharp estimate from below for the Green function of the Laplace operator. The estimate from above for the Laplacian had been proven earlier 1 Iterated Green Functions ... |

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Citation Context ...estimates, both for n = 2 and n > 2; we refer to [18]. The need of having pointwise estimates for iterated Green functions of polyharmonic operators becamesrst obvious to the authors when studying in [11] positivity questions for perturbations of the polyharmonic Dirichlet problem on the (unit-) ball B in R n : ( ) m u = f in B; u = @ @n u = = @ m 1 @n m 1 u on @B: (1.1) In that paper sharp e... |

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Citation Context ...also becomes crucial tosnd optimal pointwise estimates for the corresponding Green functions. And indeed, in recent years such estimates have been developed. Motivated by Schrodinger operators Zhao ([=-=21]-=-, see also [4]) was in 1986 thesrst to prove a sharp estimate from below for the Green function of the Laplace operator. The estimate from above for the Laplacian had been proven earlier 1 Iterated Gr... |

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Citation Context ... to prove a sharp estimate from below for the Green function of the Laplace operator. The estimate from above for the Laplacian had been proven earlier 1 Iterated Green Functions 2 in 1967 by Widman (=-=[20]-=-, see also [7, Theorem 4.6.11]). In fact, due to a result of Ancona [1], such estimates hold for quite general second order elliptic operators. For the explicit formula of those estimates, both for n ... |

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Citation Context ...d uniform. The estimates obtained here allow us to solve an open problem from [5]. This application will be treated in a separate paper [13]. 2.1 Application to coupled elliptic systems As studied in =-=[16], -=-noncooperatively coupled elliptic systems may still satisfy some positivity preserving property. For example for the system 8 : u = f " 2 v in ; v = u in ; u = v = 0 on @ ; it holds true that f ... |

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Citation Context ...11]). In fact, due to a result of Ancona [1], such estimates hold for quite general second order elliptic operators. For the explicit formula of those estimates, both for n = 2 and n > 2; we refer to =-=[18]-=-. The need of having pointwise estimates for iterated Green functions of polyharmonic operators becamesrst obvious to the authors when studying in [11] positivity questions for perturbations of the po... |

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Citation Context ...ere for K(t; x; x), i.e. on the diagonal of R n R n , they give even bounds from below. For a survey on recent results on higher order elliptic equations with emphasis on spectral theory we refer to [=-=8]-=-. 1.1 Green function estimates The starting point in [11] is an explicit expression for the Green function G m;n of (1.1), which was discovered by Boggio [3, p. 126] already at the beginning of the tw... |

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Citation Context ...k 1.3 The dependence of C k;m;n (resp. C k;m; ) on k in Theorem 1.2 can be made more explicitly. Let k 0 be the smallest number such that mk 0 1 2 n > m: Since there exist constants 0s'1;msC'1;m (see =-=-=-[5]) such that c '1;m ' 1;m (x) d (x) m C'1;m ' 1;m (x) ; Iterated Green Functions 6 where ' 1;m is a suitably normalized positivesrst eigenfunction of ( ) m under Dirichlet boundary conditions, one... |

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Citation Context ...tensively. See the book by Chung and Zhao [4]. For bounded Lipschitz domains there exists Cs1 such that for all x 2 and y 2 the estimate E x y nfyg C holds. For n = 2 Cranston and McConnell [6] even proved that C = C m(swith C an absolute constant and m(sthe Lebesgue measure of : Proof of Proposition 2.3. The proof is rather straightforward by using the expression in Lemma 2.1, Lemma 3.4 ... |

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Citation Context ...f these estimates in necessary and sucient conditions for uniform anti-maximum-principles to hold. The idea to use Green function estimates for anti-maximum-principles goes back to a paper by Takac [1=-=9-=-]. Iterated Green Functions 7 Anti-maximum-principles concern the resolvent for boundary value problems like (1.9), when the resolvent parameter is beyond thesrst eigenvalue. Usually for suciently re... |

7 |
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(Show Context)
Citation Context ... remark that if n = 2 then for any polyharmonic operator also smooth domains may be considered, which are suciently close to the disk in a suitable sense (depending on the order of the operator), see =-=[1-=-0]. Since our proofs do not distinguish between B (for m 2) or a C 1;1 -domain (for m = 1) we consider Hm;n dened on 2 with d as in (1.6). 1.2 Boundary and internal behavior The estimate (1.4) can be... |

6 |
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Citation Context ...ondition, estimates from above for the elliptic kernel are obtained by Krasovski in [15]. The higher order `heat'-kernel K(t; x; y) on R n has been considered e.g. by Davies and Barbatis in [2] and [9=-=-=-]. There for K(t; x; x), i.e. on the diagonal of R n R n , they give even bounds from below. For a survey on recent results on higher order elliptic equations with emphasis on spectral theory we refer... |

5 | The maximum principle and positive principal eigenfunctions for polyharmonic equations - Sweers - 1998 |

3 |
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(Show Context)
Citation Context ...ace operator. The estimate from above for the Laplacian had been proven earlier 1 Iterated Green Functions 2 in 1967 by Widman ([20], see also [7, Theorem 4.6.11]). In fact, due to a result of Ancona =-=[1]-=-, such estimates hold for quite general second order elliptic operators. For the explicit formula of those estimates, both for n = 2 and n > 2; we refer to [18]. The need of having pointwise estimates... |

3 | bounds on heat kernels of higher order uniformly elliptic operators
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(Show Context)
Citation Context ...undary condition, estimates from above for the elliptic kernel are obtained by Krasovski in [15]. The higher order `heat'-kernel K(t; x; y) on R n has been considered e.g. by Davies and Barbatis in [2=-=-=-] and [9]. There for K(t; x; x), i.e. on the diagonal of R n R n , they give even bounds from below. For a survey on recent results on higher order elliptic equations with emphasis on spectral theory ... |

2 |
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- Sweers
(Show Context)
Citation Context ...nterval does not depend on f the anti-maximum-principle is called uniform. The estimates obtained here allow us to solve an open problem from [5]. This application will be treated in a separate paper =-=[13-=-]. 2.1 Application to coupled elliptic systems As studied in [16], noncooperatively coupled elliptic systems may still satisfy some positivity preserving property. For example for the system 8 : u = f... |

2 |
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(Show Context)
Citation Context ...when x or y ! @ (the zero boundary condition). For more general operators, but without the eect of the boundary condition, estimates from above for the elliptic kernel are obtained by Krasovski in [15=-=-=-]. The higher order `heat'-kernel K(t; x; y) on R n has been considered e.g. by Davies and Barbatis in [2] and [9]. There for K(t; x; x), i.e. on the diagonal of R n R n , they give even bounds from b... |