The mixed problem for the Lamé system in two dimensions

K. A. Ott; R. M. Brown

doi:10.1016/j.jde.2013.03.007

The mixed problem for the Lamé system in two dimensions

K. A. Ott, R. M. Brown

Mathematics

Research output: Contribution to journal › Article › peer-review

7 Scopus citations

Abstract

We consider the mixed problem for L the Lamé system of elasticity in a bounded Lipschitz domain Ω⊂R². We suppose that the boundary is written as the union of two disjoint sets, ∂Ω=D∪N. We take traction data from the space L^p(N) and Dirichlet data from a Sobolev space W^1,p(D) and look for a solution u of Lu=0 with the given boundary conditions. In our main result, we find a scale-invariant condition on D and an exponent p₀>1 so that for 1<p<p₀, we can find a unique solution of this boundary value problem with the non-tangential maximal function of the gradient in L^p(∂Ω). We also establish the existence of a unique solution when the data is taken from Hardy spaces and Hardy-Sobolev spaces with p in (p₁, 1] for some p₁<1.

Original language	English
Pages (from-to)	4373-4400
Number of pages	28
Journal	Journal of Differential Equations
Volume	254
Issue number	12
DOIs	https://doi.org/10.1016/j.jde.2013.03.007
State	Published - Jun 15 2013

Bibliographical note

Funding Information:
E-mail address: russell.brown@uky.edu (R.M. Brown). 1 Katharine Ott is partially supported by a grant from the US National Science Foundation, DMS 1201104. 2 Russell Brown is partially supported by a grant from the Simons Foundation (#195075).

ASJC Scopus subject areas

Analysis
Applied Mathematics

Access to Document

10.1016/j.jde.2013.03.007

Cite this

@article{64b80f3a53614f07af5a461671e362ac,

title = "The mixed problem for the Lam{\'e} system in two dimensions",

abstract = "We consider the mixed problem for L the Lam{\'e} system of elasticity in a bounded Lipschitz domain Ω⊂R2. We suppose that the boundary is written as the union of two disjoint sets, ∂Ω=D∪N. We take traction data from the space Lp(N) and Dirichlet data from a Sobolev space W1,p(D) and look for a solution u of Lu=0 with the given boundary conditions. In our main result, we find a scale-invariant condition on D and an exponent p0>1 so that for 10, we can find a unique solution of this boundary value problem with the non-tangential maximal function of the gradient in Lp(∂Ω). We also establish the existence of a unique solution when the data is taken from Hardy spaces and Hardy-Sobolev spaces with p in (p1, 1] for some p1<1.",

author = "Ott, {K. A.} and Brown, {R. M.}",

note = "Funding Information: E-mail address: russell.brown@uky.edu (R.M. Brown). 1 Katharine Ott is partially supported by a grant from the US National Science Foundation, DMS 1201104. 2 Russell Brown is partially supported by a grant from the Simons Foundation (#195075).",

year = "2013",

month = jun,

day = "15",

doi = "10.1016/j.jde.2013.03.007",

language = "English",

volume = "254",

pages = "4373--4400",

number = "12",

}

TY - JOUR

T1 - The mixed problem for the Lamé system in two dimensions

AU - Ott, K. A.

AU - Brown, R. M.

N1 - Funding Information: E-mail address: russell.brown@uky.edu (R.M. Brown). 1 Katharine Ott is partially supported by a grant from the US National Science Foundation, DMS 1201104. 2 Russell Brown is partially supported by a grant from the Simons Foundation (#195075).

PY - 2013/6/15

Y1 - 2013/6/15

N2 - We consider the mixed problem for L the Lamé system of elasticity in a bounded Lipschitz domain Ω⊂R2. We suppose that the boundary is written as the union of two disjoint sets, ∂Ω=D∪N. We take traction data from the space Lp(N) and Dirichlet data from a Sobolev space W1,p(D) and look for a solution u of Lu=0 with the given boundary conditions. In our main result, we find a scale-invariant condition on D and an exponent p0>1 so that for 10, we can find a unique solution of this boundary value problem with the non-tangential maximal function of the gradient in Lp(∂Ω). We also establish the existence of a unique solution when the data is taken from Hardy spaces and Hardy-Sobolev spaces with p in (p1, 1] for some p1<1.

AB - We consider the mixed problem for L the Lamé system of elasticity in a bounded Lipschitz domain Ω⊂R2. We suppose that the boundary is written as the union of two disjoint sets, ∂Ω=D∪N. We take traction data from the space Lp(N) and Dirichlet data from a Sobolev space W1,p(D) and look for a solution u of Lu=0 with the given boundary conditions. In our main result, we find a scale-invariant condition on D and an exponent p0>1 so that for 10, we can find a unique solution of this boundary value problem with the non-tangential maximal function of the gradient in Lp(∂Ω). We also establish the existence of a unique solution when the data is taken from Hardy spaces and Hardy-Sobolev spaces with p in (p1, 1] for some p1<1.

UR - http://www.scopus.com/inward/record.url?scp=84876195397&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84876195397&partnerID=8YFLogxK

U2 - 10.1016/j.jde.2013.03.007

DO - 10.1016/j.jde.2013.03.007

M3 - Article

AN - SCOPUS:84876195397

VL - 254

SP - 4373

EP - 4400

IS - 12

ER -

The mixed problem for the Lamé system in two dimensions

Abstract

Bibliographical note

ASJC Scopus subject areas

Access to Document

Other files and links

Fingerprint

Cite this