TY  - JOUR
A1  - Dines, Nicoleta
A1  - Liu, Xiaochun
A1  - Schulze, Bert-Wolfgang
T1  - Edge quantisation of elliptic operators
JF  - Preprint / Universität Potsdam, Institut für Mathematik, Arbeitsgruppe Partiell
N2  - The ellipticity of operators on a manifold with edge is defined as the bijectivity of the components of a principal symbolic hierarchy sigma = (sigma(psi), sigma(boolean AND)), where the second component takes values in operators on the infinite model cone of the local wedges. In the general understanding of edge problems there are two basic aspects: Quantisation of edge-degenerate operators in weighted Sobolev spaces, and verifying the ellipticity of the principal edge symbol sigma(boolean AND) which includes the (in general not explicity known) number of additional conditions of trace and potential type on the edge. We focus here on these questions and give explicit answers for a wide class of elliptic operators that are connected with the ellipticity of edge boundary value problems and reductions to the boundary. In particular, we study the edge quantisation and ellipticity for Dirichlet-Neumann operators with respect to interfaces of some codimension on a boundary. We show analogues of the Agranovich-Dynin formula for edge boundary value problems.
Y1  - 2009
UR  - http://www.springerlink.com/content/103082
U6  - https://doi.org/10.1007/s00605-008-0058-y
SN  - 1437-739X
ER  - 
TY  - JOUR
A1  - Liu, Xiaochun
A1  - Jahn, Bor-Ming
A1  - Dong, Shuwen
A1  - Li, Huimin
A1  - Oberhänsli, Roland
T1  - Neoproterozoic granitoid did not record ultrahigh-pressure metamorphism from the Southern Dabieshan of China
N2  - It has been often debated whether all granitic gneisses associated with coesite-bearing eclogites in southern Dabieshan, China, have also been subjected to ultrahigh-pressure (UHP) metamorphism. We show here that a metagranitoid adjacent to the Bixiling eclogite-ultramafic complex has preserved primary granitic textures and an igneous mineral assemblage of biotite + plagioclase + microcline + quartz + allanite +/- amphibole. The absence of UPH recrystallization for the metagranitoid is particularly manifested by the conservation of euhedral-zoned plagioclase phenocrysts, the lack of corona garnets around igneous biotite, and the presence of an igneous mineral assemblage in zircons. The only metamorphic overprint was the epidote-amphibolite facies metamorphism characterized by the assemblage of biotite + phengiticmica + epidote + albite + K-feldspar + quartz +/- amphibole Metamorphic conditions are estimated at ca. 550degrees-680degreesC and 6-13 kbar for the metagranitoid and its amphibolitic enclave. Geochemically, the metagranitoid is similar to its country gneiss and shows an affinity to volcanic arc granitoid. Zircon U-Pb dating suggests that the Bixiling metagranitoid was emplaced during the Neoproterozoic (729+/-4 Ma), when most other granitic rocks and the protoliths of eclogite were also formed in Dabieshan. Taking into account the discovery of non-UHP granitic gneisses in other places, we argue that part of Neoproterozoic granitic rocks in the Dabieshan and Sulu terranes have escaped UHP metamorphism during the Triassic deep subduction of the continental crust as a consequence of a lack of penetrative deformation and fluid-rock interaction
Y1  - 2003
SN  - 0022-1376
ER  - 
TY  - JOUR
A1  - Liu, Xiaochun
A1  - Schulze, Bert-Wolfgang
T1  - Ellipticity on manifolds with edges and boundary
N2  - Ellipticity of a manifold with edges and boundary is connected to boundary and edge conditions that complete corresponding operators to Fredholm operators between weighted Sobolev spaces. We study a new parameter-dependent calculus of elliptic operators, where the interior symbols have specific properties on the boundary. We construct elliptic operators with a prescribed number of edge conditions and obtain isomorphisms in the scale of edge Sobolev spaces
Y1  - 2005
SN  - 0026-9255
ER  - 
TY  - JOUR
A1  - Liu, Xiaochun
A1  - Witt, Ingo
T1  - Pseudodifferential calculi on the half-line respecting prescribed asymptotic types
N2  - Given asymptotics types P, Q, pseudodifferential operators A is an element of L-cl(mu) (R+) are constructed in such a way that if u(t) possesses conormal asymptotics of type P as t --> +0, then Au(t) possesses conormal asymptotics of type Q as t --> +0. This is achieved by choosing the operators A in Schulze's cone algebra on the half-line R+, controlling their complete Mellin symbols {sigma(M)(u-j) (A); j is an element of N}, and prescribing the mapping properties of the residual Green operators. The constructions lead to a coordinate invariant calculus, including trace and potential operators at t = 0, in which a parametrix construction for the elliptic elements is possible. Boutet de Monvel's calculus for pseudodifferential boundary problems occurs as a special case when P = Q is the type resulting from Taylor expansion at t = 0.
Y1  - 2004
SN  - 0378-620X
ER  -