### Abstract

Given a bounded Lipschitz domain Ω ⊂ ℝ^{n} n ≥ 3, we prove that the Poisson's problem for the Laplacian with right-hand side in L_{-t}^{p}(Ω), Robin-type boundary datum in the Besov space B_{p}^{1-1/p-t,p}(∂Ω) and non-negative, non-everywhere vanishing Robin coefficient b ∈ L ^{n-1}(∂ω), is uniquely solvable in the class L _{2-t}^{p}(Omega;) for (t, 1/p) ∈ ν_{ε}, where ν_{ε} (ε ≥ 0) is an open (Ω,b)-dependent plane region and ν_{0} is to be interpreted ad the common (optimal) solvability region for all Lipschitz domains. We prove a similar regularity result for the Poisson's problem for the 3-dimensional Lamé System with traction-type Robin boundary condition. All solutions are expressed as boundary layer potentials.

Original language | English (US) |
---|---|

Pages (from-to) | 181-204 |

Number of pages | 24 |

Journal | Revista Matematica Iberoamericana |

Volume | 22 |

Issue number | 1 |

State | Published - 2006 |

Externally published | Yes |

### Fingerprint

### Keywords

- Besov spaces
- Boundary layer potentials
- Lamé system
- Non-smooth domains
- Poisson's problem
- Regularity of PDE's
- Robin condition
- TYiebel-Lizorkin spaces

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Revista Matematica Iberoamericana*,

*22*(1), 181-204.

**The Poisson's problem for the Laplacian with Robin boundary condition in non-smooth domains.** / Lanzani, Loredana; Méndez, Osvaldo.

Research output: Contribution to journal › Article

*Revista Matematica Iberoamericana*, vol. 22, no. 1, pp. 181-204.

}

TY - JOUR

T1 - The Poisson's problem for the Laplacian with Robin boundary condition in non-smooth domains

AU - Lanzani, Loredana

AU - Méndez, Osvaldo

PY - 2006

Y1 - 2006

N2 - Given a bounded Lipschitz domain Ω ⊂ ℝn n ≥ 3, we prove that the Poisson's problem for the Laplacian with right-hand side in L-tp(Ω), Robin-type boundary datum in the Besov space Bp1-1/p-t,p(∂Ω) and non-negative, non-everywhere vanishing Robin coefficient b ∈ L n-1(∂ω), is uniquely solvable in the class L 2-tp(Omega;) for (t, 1/p) ∈ νε, where νε (ε ≥ 0) is an open (Ω,b)-dependent plane region and ν0 is to be interpreted ad the common (optimal) solvability region for all Lipschitz domains. We prove a similar regularity result for the Poisson's problem for the 3-dimensional Lamé System with traction-type Robin boundary condition. All solutions are expressed as boundary layer potentials.

AB - Given a bounded Lipschitz domain Ω ⊂ ℝn n ≥ 3, we prove that the Poisson's problem for the Laplacian with right-hand side in L-tp(Ω), Robin-type boundary datum in the Besov space Bp1-1/p-t,p(∂Ω) and non-negative, non-everywhere vanishing Robin coefficient b ∈ L n-1(∂ω), is uniquely solvable in the class L 2-tp(Omega;) for (t, 1/p) ∈ νε, where νε (ε ≥ 0) is an open (Ω,b)-dependent plane region and ν0 is to be interpreted ad the common (optimal) solvability region for all Lipschitz domains. We prove a similar regularity result for the Poisson's problem for the 3-dimensional Lamé System with traction-type Robin boundary condition. All solutions are expressed as boundary layer potentials.

KW - Besov spaces

KW - Boundary layer potentials

KW - Lamé system

KW - Non-smooth domains

KW - Poisson's problem

KW - Regularity of PDE's

KW - Robin condition

KW - TYiebel-Lizorkin spaces

UR - http://www.scopus.com/inward/record.url?scp=33745096271&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33745096271&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:33745096271

VL - 22

SP - 181

EP - 204

JO - Revista Matematica Iberoamericana

JF - Revista Matematica Iberoamericana

SN - 0213-2230

IS - 1

ER -