Abstract
We propose the analogues of boundary layer potentials for the sub-Laplacian on homogeneous Carnot groups/stratified Lie groups and prove continuity results for them. In particular, we show continuity of the single layer potential and establish the Plemelj type jump relations for the double layer potential. We prove sub-Laplacian adapted versions of the Stokes theorem as well as of Green's first and second formulae on homogeneous Carnot groups. Several applications to boundary value problems are given. As another consequence, we derive formulae for traces of the Newton potential for the sub-Laplacian to piecewise smooth surfaces. Using this we construct and study a nonlocal boundary value problem for the sub-Laplacian extending to the setting of the homogeneous Carnot groups M. Kac's “principle of not feeling the boundary”. We also obtain similar results for higher powers of the sub-Laplacian. Finally, as another application, we prove refined versions of Hardy's inequality and of the uncertainty principle.
| Original language | English |
|---|---|
| Pages (from-to) | 483-528 |
| Number of pages | 46 |
| Journal | Advances in Mathematics |
| Volume | 308 |
| DOIs | |
| Publication status | Published - Feb 21 2017 |
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Keywords
- Hardy inequality
- Homogeneous Carnot group
- Integral boundary condition
- Layer potentials
- Newton potential
- Stratified group
- Sub-Laplacian
ASJC Scopus subject areas
- Mathematics(all)
Cite this
Layer potentials, Kac's problem, and refined Hardy inequality on homogeneous Carnot groups. / Ruzhansky, Michael; Suragan, Durvudkhan.
In: Advances in Mathematics, Vol. 308, 21.02.2017, p. 483-528.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Layer potentials, Kac's problem, and refined Hardy inequality on homogeneous Carnot groups
AU - Ruzhansky, Michael
AU - Suragan, Durvudkhan
PY - 2017/2/21
Y1 - 2017/2/21
N2 - We propose the analogues of boundary layer potentials for the sub-Laplacian on homogeneous Carnot groups/stratified Lie groups and prove continuity results for them. In particular, we show continuity of the single layer potential and establish the Plemelj type jump relations for the double layer potential. We prove sub-Laplacian adapted versions of the Stokes theorem as well as of Green's first and second formulae on homogeneous Carnot groups. Several applications to boundary value problems are given. As another consequence, we derive formulae for traces of the Newton potential for the sub-Laplacian to piecewise smooth surfaces. Using this we construct and study a nonlocal boundary value problem for the sub-Laplacian extending to the setting of the homogeneous Carnot groups M. Kac's “principle of not feeling the boundary”. We also obtain similar results for higher powers of the sub-Laplacian. Finally, as another application, we prove refined versions of Hardy's inequality and of the uncertainty principle.
AB - We propose the analogues of boundary layer potentials for the sub-Laplacian on homogeneous Carnot groups/stratified Lie groups and prove continuity results for them. In particular, we show continuity of the single layer potential and establish the Plemelj type jump relations for the double layer potential. We prove sub-Laplacian adapted versions of the Stokes theorem as well as of Green's first and second formulae on homogeneous Carnot groups. Several applications to boundary value problems are given. As another consequence, we derive formulae for traces of the Newton potential for the sub-Laplacian to piecewise smooth surfaces. Using this we construct and study a nonlocal boundary value problem for the sub-Laplacian extending to the setting of the homogeneous Carnot groups M. Kac's “principle of not feeling the boundary”. We also obtain similar results for higher powers of the sub-Laplacian. Finally, as another application, we prove refined versions of Hardy's inequality and of the uncertainty principle.
KW - Hardy inequality
KW - Homogeneous Carnot group
KW - Integral boundary condition
KW - Layer potentials
KW - Newton potential
KW - Stratified group
KW - Sub-Laplacian
UR - http://www.scopus.com/inward/record.url?scp=85007505729&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85007505729&partnerID=8YFLogxK
U2 - 10.1016/j.aim.2016.12.013
DO - 10.1016/j.aim.2016.12.013
M3 - Article
VL - 308
SP - 483
EP - 528
JO - Advances in Mathematics
JF - Advances in Mathematics
SN - 0001-8708
ER -