ساره علي خليل

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الأبحاث

Corner Operators with Symbol Hierarchies

Authors : D. -C. Chang, S. Khalil, and B.-W. Schulze

Abstract :This paper outlines an approach for studying operators on stratiﬁed spaces M ∈ Mk with regular singularities of higher order k. Smoothness corresponds to k = 0. Manifolds with smooth boundaries belong to the category M1. The case k = 1 generally indicates conical or edge singularities. Boutet de Monvel’s algebra of boundary value problems (BVPs) with the transmission property at the boundary may be interpreted as a special singular operator calculus for k = 1. Also, BVPs A with violated transmission properties belong to edge calculus and are controlled by pairs {σj(A)}j=0,1, consisting of interior and boundary symbols. Singularities of M ∈ Mk for higher order k give rise to a sequence of strata s(M) = {sj(M)}j=0,..,k, where sj(M) ∈ M0. Operators A in corresponding algebras of operators (corner-degenerate in stretched variables) are determined by a hierarchy of symbols σ(A) = {σj(A)}j=0,..,k, modulo lower order terms. Those express ellipticity and parametrices A(−1) in weighted corner Sobolev spaces, containing sequences of real weights γj. Components σj(A) for j > 0, depending on variables and covariables in T∗(sj(M))\0, act as operator families on inﬁnite straight cones with compact singular links in Mj−1, and σ0(A) is the standard principal symbol on T∗(s0(M))\0.

Keywords :Boutet de Monvel’s calculus, pseudo-diﬀerential operators, stratiﬁed spaces, singular cones, Mellin symbols with values in the edge calculus, Kegel space

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Analysis on regular corner spaces

Authors :D. -C. Chang, S. Khalil, and B.-W. Schulze

Abstract :We establish a new approach of treating elliptic boundary value problems (BVPs) on manifolds with boundary and regular corners, up to singularity order 2. Ellipticity and parametrices are obtained in terms of symbols taking values in algebras of BVPs on manifolds of corresponding lower singularity orders. Those refer to Boutet de Monvel’s calculus of operators with the transmission property, see Boutet de Monvel (Acta Math 126:11–51, 1971) for the case of smooth boundary. On corner configuration operators act in spaces with multiple weights. We mainly study the case of upper left entries in the respective 2x2- operator block-matrices of such a calculus. Green operators in the sense of Boutet de Monvel (Acta Math 126:11–51, 1971) analogously appear in singular cases, and they are complemented by contributions of Mellin type. We formulate a result on ellipticity and the Fredholm property in weighted corner spaces, with parametrices of analogous kind

Keywords :Boutet de Monvel’s calculus, pseudo-diﬀerential operators, Mellin symbols with values in the edge calculus, Kegel space, Singular cones

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Structures for Corner Boundary Value Problems

Authors :S. Khalil, B.-W. Schulze

Abstract :Boutet de Monvel’s calculus [1] of pseudo-differential boundary value problems (BVPs) may be extended from the smooth case to corner manifolds with boundary. We outline here the approach of [7] which is to some extent parallel to the corner analysis off the boundary. Here we refer to new progress from [42]

Keywords :Mellin operators, corner-degenerate boundary value problems, symbol hierarchies, Kegel spaces, weighted corner spaces

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Weighted Corner Spaces

Authors : D. -C. Chang, S. Khalil, and B.-W. Schulze

Abstract :This article contains new elements of the pseudo-diﬀerential analysis on speciﬁc classes of stratiﬁed spaces. We study structures appearing in parametrices of elliptic operators and we establish the interplay between corner degenerate symbols of operators and weighted Sobolev spaces on the underlying spaces M with the singular geometry of cones, edges or corners. In particular, we establish an accessible approach to corner Sobolev spaces. On the smooth part of M the pseudo-diﬀerential operators locally refer to the Fourier transform. Close to the singularities we employ diﬀerent variants of Mellin pseudo-diﬀerential operators, in order to control phenomena up to the singularities

Keywords :pseudo-diﬀerential operators, elliptic operators, parametrices of elliptic operator Mellin pseudo-diﬀerential operators, weighted Sobolev spaces

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Calculus on a Manifold with Edge and Boundary

Authors :S. Khalil, and B.-W. Schulze

Abstract :We study elements of the calculus of boundary value problems in a variant of Boutet de Monvel’s algebra [1] on a manifold N with edge and boundary. If the boundary is empty then the approach corresponds to [26] and other papers from the subsequent development. For non-trivial boundary we study Mellin-edge quantizations and compositions within the structure in terms a new Mellin-edge quantization, compared with a more traditional technique. Similar structures in the closed case have been studied in [10]

Keywords : pseudo-diﬀerential operators, Manifold with edge, Boutet de Monvel’s calculus, Mellin quantization

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Boundary problems on a manifold with edge

Authors : S. Khalil, and B.-W. Schulze

Abstract :We establish a calculus of boundary value problems (BVPs) on a manifold N with boundary and edge, based on Boutet de Monvel’s theory of BVPs in the case of a smooth boundary and on the edge calculus, where in the present case the model cone has a base which is a compact manifold with boundary. The corresponding calculus with boundary and edge is a unification of both structures and controls different operator-valued symbolic structures, in order to obtain ellipticity and parametrices

Keywords : Manifolds with singularities, Boutet de Monvel’s calculus, operators on manifolds with edge and boundary, distribution with asymptotics, ellipticity, Fredholm property

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Boundary Value Problem in Boutet de Monvel's Calculus on Manifolds with Edg

Authors :S. Khalil, and B.-W. Schulze

Abstract :The present exposition is a contribution to the third conference AMINSE. Our topic is related to research teams from the I. Vekua Institute of Applied Mathematics and Iv. Javakhishvili Tbilisi State University who are working in the tradition of I. Vekua. The authors are happy to give an overview of in memory of Professor David Gordeziani who contributed so much for building up mathematical institutions in Tbilisi to attractive centers of research in many areas, including singular analysis, with members Jaiani, Chinchaldze, Duduchava, Natroshvili and many others

Keywords : pseudo-diﬀerential operators, Boutet de Monvel’s calculus on manifolds with edge, Partial differential operator

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On Generalized Simultaneous Nearest Point in Normed Spaces

Authors : M. Rawashdeh, S. AL-sharif, and S. Khalil

Abstract :In this paper, we study the problem of generalized simultaneous approximation in terms of the Minkowski functional. We develop a theory of generalized best simultaneous approximation in quotient spaces and introduce equivalent assertions between the subspaces W and W+M and the quotient space W/M. Some other results regarding generalized simultaneous approximation in Banach space are presented

Keywords :Generalized nearest point , ‎normed spaces , Simultaneous approximation

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New Generalization of f- Best Simultaneous Approximation in Topological Vector Spaces

Authors : M. Rawashdeh and S. Khalil

Abstract :Let 𝐾 be a nonempty subset of a Hausdorff topological vector space 𝑋, and let 𝑓 be a real-valued continuous function on 𝑋. If for each 𝑥 = (𝑥1, 𝑥2,...,𝑥𝑛)∈𝑋𝑛 , there exists 𝑘0 ∈ 𝐾 such that 𝐹𝐾(𝑥) = ∑𝑛 𝑖=1 𝑓(𝑥𝑖 − 𝑘0) = inf{∑𝑛 𝑖=1 𝑓(𝑥𝑖 − 𝑘) : 𝑘 ∈ 𝐾}, then 𝐾 is called 𝑓-simultaneously proximal and 𝑘0 is called 𝑓-best simultaneous approximation for 𝑥 in 𝐾. In this paper, we study the problem of 𝑓-simultaneous approximation for a vector subspace 𝐾 in 𝑋. Some other results regarding 𝑓-simultaneous approximation in quotient space are presented

Keywords :Generalized nearest point , ‎normed spaces , Simultaneous approximation

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