scholarly journals Elliptic Theory on Manifolds with Corners: II. Homotopy Classification and K-Homology

2008 ◽  
pp. 207-226 ◽  
Author(s):  
Vladimir Nazaikinskii ◽  
Anton Savin ◽  
Boris Sternin
Keyword(s):  
Homotopy Classification ◽  
Elliptic Theory ◽  
Manifolds With Corners

On the homotopy classification of elliptic operators on manifolds with corners

2007 ◽  
Vol 75 (2) ◽  
pp. 186-189 ◽  
Author(s):  
V. E. Nazaikinskii ◽  
A. Yu. Savin ◽  
B. Yu. Sternin
Keyword(s):  
Elliptic Operators ◽  
Homotopy Classification ◽  
Manifolds With Corners

scholarly journals A signature formula for manifolds with corners of codimension two

Topology ◽  
1997 ◽  
Vol 36 (5) ◽  
pp. 1055-1075 ◽  
Author(s):  
Andrew Hassell ◽  
Rafe Mazzeo ◽  
Richard B. Melrose
Keyword(s):  
Codimension Two ◽  
Manifolds With Corners ◽  
Signature Formula

scholarly journals Elliptic Theory for Sets with Higher Co-dimensional Boundaries

2021 ◽  
Vol 274 (1346) ◽  
Author(s):  
G. David ◽  
J. Feneuil ◽  
S. Mayboroda
Keyword(s):  
Green Function ◽  
Elliptic Operator ◽  
Harmonic Measure ◽  
Lipschitz Constant ◽  
Extension Theorems ◽  
Content Type ◽  
Elliptic Theory ◽  
Basic Set ◽  
Degenerate Elliptic ◽  
The Green Function

Many geometric and analytic properties of sets hinge on the properties of elliptic measure, notoriously missing for sets of higher co-dimension. The aim of this manuscript is to develop a version of elliptic theory, associated to a linear PDE, which ultimately yields a notion analogous to that of the harmonic measure, for sets of codimension higher than 1. To this end, we turn to degenerate elliptic equations. Let Γ ⊂ R n \Gamma \subset \mathbb {R}^n be an Ahlfors regular set of dimension d > n − 1 d>n-1 (not necessarily integer) and Ω = R n ∖ Γ \Omega = \mathbb {R}^n \setminus \Gamma . Let L = − div ⁡ A ∇ L = - \operatorname {div} A\nabla be a degenerate elliptic operator with measurable coefficients such that the ellipticity constants of the matrix A A are bounded from above and below by a multiple of dist ⁡ ( ⋅ , Γ ) d + 1 − n \operatorname {dist}(\cdot , \Gamma )^{d+1-n} . We define weak solutions; prove trace and extension theorems in suitable weighted Sobolev spaces; establish the maximum principle, De Giorgi-Nash-Moser estimates, the Harnack inequality, the Hölder continuity of solutions (inside and at the boundary). We define the Green function and provide the basic set of pointwise and/or L p L^p estimates for the Green function and for its gradient. With this at hand, we define harmonic measure associated to L L , establish its doubling property, non-degeneracy, change-of-the-pole formulas, and, finally, the comparison principle for local solutions. In another article to appear, we will prove that when Γ \Gamma is the graph of a Lipschitz function with small Lipschitz constant, we can find an elliptic operator L L for which the harmonic measure given here is absolutely continuous with respect to the d d -Hausdorff measure on Γ \Gamma and vice versa. It thus extends Dahlberg’s theorem to some sets of codimension higher than 1.

scholarly journals Homotopy classification of elliptic operators on stratified manifolds

2006 ◽  
Vol 73 (3) ◽  
pp. 407-411 ◽  
Author(s):  
V. E. Nazaikinskii ◽  
A. Yu. Savin ◽  
B. Yu. Sternin
Keyword(s):  
Elliptic Operators ◽  
Homotopy Classification
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