## Estimates for Green Function and the Poisson Kernels of higher order Dirichlet boundary value problems

Venue: | J. Differential Equations |

Citations: | 15 - 2 self |

### BibTeX

@ARTICLE{Sweers_estimatesfor,

author = {Guido Sweers},

title = {Estimates for Green Function and the Poisson Kernels of higher order Dirichlet boundary value problems},

journal = {J. Differential Equations},

year = {},

volume = {205},

pages = {466--487}

}

### OpenURL

### Abstract

Pointwise estimates are derived for the kernels associated to the polyharmonic Dirichlet problem on bounded smooth domains. As a consequence one obtains optimal weighted L p-L q-regularity estimates for weights involving the distance function.

### Citations

2083 |
Trudinger N., Elliptic Partial Differential Equations of Second Order, 2nd edition
- Gilbarg
- 1983
(Show Context)
Citation Context ... (−∆) m u = f in Ω, ⎪⎩ ( ∂ ∂ν )k u = 0 on ∂Ω with 0 ≤ k ≤ m − 1, (14) where Ω ⊂ Rn is bounded and has the boundary regularity as before. First we recall an estimate involving the Riesz potential (see =-=[8]-=-). Defining Kγ(x) = |x| −γ and � (Kγ ∗ f) (x) := |x − y| −γ f(y)dy, one has: Lemma 16 Let Ω ⊂ R n be bounded, γ < n and 1 ≤ p ≤ q ≤ ∞. If γ 1 + 1 q − 1 p then there is Cn−γr,Ω > 0 such that for all f ... |

566 | Non-homogeneous boundary value problems and applications - Lions, Magenes - 1972 |

354 |
Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions
- Agmon, Douglis, et al.
(Show Context)
Citation Context ...ingularity x = y and such that it has a length of the same magnitude as d(x). In the following lemma we state the existence of such a path. Lemma 7 Let x ∈ Ω and y ∈ ¯ Ω. There exists a curve γ y x : =-=[0, 1]-=- → ¯ Ω with γ y x(0) = x, γ y x(1) ∈ ∂Ω and such that: (1) for every t ∈ [0, 1] : |γy x(t) − y| ≥ 1 |x − y| , 2 (2) l ≤ (1 + π) d(x) where l is the length of γy x. Moreover, letting ˜γ y x : [0, l] → ... |

53 |
Sulle funzioni di Green d’ordine m
- Boggio
- 1905
(Show Context)
Citation Context ...ved the estimates for the general case. In [9] the estimates as in Theorem 3 are given for the case that Ω is a ball in R n . There the authors could use the explicit formula of Gm given by Boggio in =-=[3]-=-. For balls the Green function associated to problem (1) is positive. 4sFor general domains one cannot expect an explicit formula and instead we will proceed by the estimates of Krasovskiĭ for Gm and ... |

38 | Positivity for equations involving polyharmonic operators with Dirichlet boundary conditions
- Grunau, Sweers
- 1997
(Show Context)
Citation Context ...rp (see e.g. [11]). 2 � �sFor m = 1 or m ≥ 2 and Ω = B it is known that the Green function is positive and can even be estimated from below by a positive function with the same singular behavior (see =-=[9]-=-). Let us remind the reader that for m ≥ 2 the Green function in general is not positive. We believe however that for general domains the optimal behavior in absolute values is captured in our estimat... |

28 |
Parabolic Systems
- Eidel’man
- 1969
(Show Context)
Citation Context ...e available. Barbatis [2] considered higher order parabolic problems on domains and derived pointwise estimates for the kernel using a non-Euclidean metric. Classical estimates by Eidel’man (see e.g. =-=[6]-=-) for higher order parabolic systems do not consider domains with boundary. For a survey on spectral theory of higher order elliptic operators, including some estimates for the corresponding kernels, ... |

14 | Lp spectral theory of higher-order elliptic differential operators
- Davies
- 1997
(Show Context)
Citation Context ... order parabolic systems do not consider domains with boundary. For a survey on spectral theory of higher order elliptic operators, including some estimates for the corresponding kernels, we refer to =-=[5]-=-. Finally we would like to remark that we do not pretend that our pointwise estimates are completely new. However we have not been able to find any reference to such estimates for the special type of ... |

10 | Sharp estimates for iterated Green function
- Grunau
(Show Context)
Citation Context ...stimates that contain growth rates near the boundary. These estimates seem to be optimal and indeed, when we consider Gm for Ω = B a ball in R n the growth rates near the boundary are sharp (see e.g. =-=[11]-=-). 2 � �sFor m = 1 or m ≥ 2 and Ω = B it is known that the Green function is positive and can even be estimated from below by a positive function with the same singular behavior (see [9]). Let us remi... |

8 |
Isolation of Singularities of the Green Function
- Krasovskiĭ
- 1967
(Show Context)
Citation Context ...ing by their own merits. A special case for m = 1 appears in [7]. Not only we will derive estimates for those kernels but also for their derivatives. The main tool will be the result of Krasovskiĭ in =-=[12]-=- where he considered general elliptic operators and boundary conditions. The estimates he derived did not involve special growth rates near the boundary. We instead will focus on estimates that contai... |

7 |
and nonlinear heat equations in L q δ spaces and universal bounds for global solutions
- Fila, Souplet, et al.
- 2001
(Show Context)
Citation Context ...1 (Ω) These kinds of estimates, for general m and n, and also L p -L q estimates will be addressed in Section 4. The estimates are interesting by their own merits. A special case for m = 1 appears in =-=[7]-=-. Not only we will derive estimates for those kernels but also for their derivatives. The main tool will be the result of Krasovskiĭ in [12] where he considered general elliptic operators and boundary... |

6 | Uniform anti-maximum principles
- Clément
(Show Context)
Citation Context ...1 1 − p q < min � 2m, 1� appears with n a strict inequality. Such estimates will also follow through regularity results in Lp , Poincaré estimates, Sobolev imbeddings and dual Sobolev imbeddings. See =-=[4]-=-. Remark 21 In a similar way one may also derive estimates for combinations of boundary behavior and derivatives. For example if n = m = 2 one finds with θ ∈ [0, 1] : � � �d(.) −1+2θ Dxu � � �L ≤ C ∞ ... |

5 | heat kernel bounds and Finsler-type - Barbatis, Sharp |

4 | The role of positive boundary data in the generalized clamped plate equation, ZAMP 49
- Grunau
- 1998
(Show Context)
Citation Context ... is not positive. We believe however that for general domains the optimal behavior in absolute values is captured in our estimates. Sharp estimates for Km−1 and Km−2 in case of a ball can be found in =-=[10]-=-. Instead of using Krasovskiĭ’s result one might use appropriate “heat kernel” estimates. Indeed, integrating pointwise estimates for the parabolic kernel p(t, x, y) with respect to t from 0 to ∞, yie... |