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A. B. Tumpach - One of the best experts on this subject based on the ideXlab platform.
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Banach Poisson–Lie Groups and Bruhat–Poisson Structure of the Restricted Grassmannian
Communications in Mathematical Physics, 2020Co-Authors: A. B. TumpachAbstract:The first part of this paper is devoted to the theory of Poisson–Lie groups in the Banach setting. Our starting point is the straightforward adaptation of the notion of Manin triples to the Banach context. The difference with the finite-dimensional case lies in the fact that a Duality Pairing between two non-reflexive Banach spaces is necessary weak (as opposed to a strong Pairing where one Banach space can be identified with the dual space of the other). The notion of generalized Banach Poisson manifolds introduced in this paper is compatible with weak Duality Pairings between the tangent space and a subspace of the dual. We investigate related notion like Banach Lie bialgebras and Banach Poisson–Lie groups, suitably generalized to the non-reflexive Banach context. The second part of the paper is devoted to the treatment of particular examples of Banach Poisson–Lie groups related to the restricted Grassmannian and the KdV hierarchy. More precisely, we construct a Banach Poisson–Lie group structure on the unitary restricted Banach Lie group which acts transitively on the restricted Grassmannian. A“dual” Banach Lie group consisting of (a class of) upper triangular bounded operators admits also a Banach Poisson–Lie group structure of the same kind. We show that the restricted Grassmannian inherits a generalized Banach Poisson structure from the unitary Banach Lie group, called Bruhat–Poisson structure. Moreover the action of the triangular Banach Poisson–Lie group on it is a Poisson map. This action generates the KdV hierarchy, and its orbits are the Schubert cells of the restricted Grassmannian.
Wontae Kim - One of the best experts on this subject based on the ideXlab platform.
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an existence result for nonhomogeneous quasilinear parabolic equations beyond the Duality Pairing
arXiv: Analysis of PDEs, 2019Co-Authors: Karthik Adimurthi, Sunsig Byun, Wontae KimAbstract:In this paper, we prove existence of \emph{very weak solutions} to nonhomogeneous quasilinear parabolic equations beyond the Duality Pairing. The main ingredients are a priori esitmates in suitable weighted spaces combined with the compactness argument developed in \cite{bulicek2018well}. In order to obtain the a priori estimates, we make use of the full Calder\'on-Zygmund machinery developed in the past few years and combine it with some sharp bounds for the subclass of Muckenhoupt weights considered in this paper.
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partial existence result for homogeneous quasilinear parabolic problems beyond the Duality Pairing
arXiv: Analysis of PDEs, 2018Co-Authors: Karthik Adimurthi, Sunsig Byun, Wontae KimAbstract:In this paper, we study the existence of distributional solutions solving \cref{main-3} on a bounded domain $\Omega$ satisfying a uniform capacity density condition where the nonlinear structure $\mathcal{A}(x,t,\nabla u)$ is modelled after the standard parabolic $p$-Laplace operator. In this regard, we need to prove a priori estimates for the gradient of the solution below the natural exponent and a higher integrability result for very weak solutions at the initial boundary. The elliptic counterpart to these two estimates are fairly well developed over the past few decades, but no analogous theory exists in the quasilinear parabolic setting. Two important features of the estimates proved here are that they are non-perturbative in nature and we are able to take non-zero boundary data. \emph{As a consequence, our estimates are new even for the heat equation on bounded domains.} This partial existence result is a nontrivial extension of the existence theory of very weak solutions from the elliptic setting to the quasilinear parabolic setting. Even though we only prove partial existence result, nevertheless we establish the necessary framework that when proved would lead to obtaining the full result for the homogeneous problem.
Karthik Adimurthi - One of the best experts on this subject based on the ideXlab platform.
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an existence result for nonhomogeneous quasilinear parabolic equations beyond the Duality Pairing
arXiv: Analysis of PDEs, 2019Co-Authors: Karthik Adimurthi, Sunsig Byun, Wontae KimAbstract:In this paper, we prove existence of \emph{very weak solutions} to nonhomogeneous quasilinear parabolic equations beyond the Duality Pairing. The main ingredients are a priori esitmates in suitable weighted spaces combined with the compactness argument developed in \cite{bulicek2018well}. In order to obtain the a priori estimates, we make use of the full Calder\'on-Zygmund machinery developed in the past few years and combine it with some sharp bounds for the subclass of Muckenhoupt weights considered in this paper.
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partial existence result for homogeneous quasilinear parabolic problems beyond the Duality Pairing
arXiv: Analysis of PDEs, 2018Co-Authors: Karthik Adimurthi, Sunsig Byun, Wontae KimAbstract:In this paper, we study the existence of distributional solutions solving \cref{main-3} on a bounded domain $\Omega$ satisfying a uniform capacity density condition where the nonlinear structure $\mathcal{A}(x,t,\nabla u)$ is modelled after the standard parabolic $p$-Laplace operator. In this regard, we need to prove a priori estimates for the gradient of the solution below the natural exponent and a higher integrability result for very weak solutions at the initial boundary. The elliptic counterpart to these two estimates are fairly well developed over the past few decades, but no analogous theory exists in the quasilinear parabolic setting. Two important features of the estimates proved here are that they are non-perturbative in nature and we are able to take non-zero boundary data. \emph{As a consequence, our estimates are new even for the heat equation on bounded domains.} This partial existence result is a nontrivial extension of the existence theory of very weak solutions from the elliptic setting to the quasilinear parabolic setting. Even though we only prove partial existence result, nevertheless we establish the necessary framework that when proved would lead to obtaining the full result for the homogeneous problem.
Thierry Mignon - One of the best experts on this subject based on the ideXlab platform.
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Quantum Serre theorem as a Duality between quantum D-modules
'Oxford University Press (OUP)', 2016Co-Authors: Hiroshi Iritani, Etienne Mann, Thierry MignonAbstract:We give an interpretation of quantum Serre of Coates and Givental as a Duality of twisted quantum D-modules. This interpretation admits a non-equivariant limit, and we obtain a precise relationship among (1) the quantum D-module of X twisted by a convex vector bundle E and the Euler class, (2) the quantum D-module of the total space of the dual bundle E ∨ → X, and (3) the quantum D-module of a submanifold Z ⊂ X cut out by a regular section of E. When E is the anticanonical line bundle K −1 X , we identify these twisted quantum D-modules with second structure connections with different parameters, which arise as Fourier-Laplace transforms of the quantum D-module of X. In this case, we show that the Duality Pairing is identified with Dubrovin\u27s second metric. CONTENT
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QUANTUM SERRE IN TERMS OF QUANTUM D-MODULES
2014Co-Authors: Hiroshi Iritani, Etienne Mann, Thierry MignonAbstract:We give an interpretation of quantum Serre of Coates and Givental as a Duality of twisted quantum D-modules. This interpretation admits a non-equivariant limit, and we obtain a precise relationship among (1) the quantum D-module of X twisted by a convex vector bundle E and the Euler class, (2) the quantum D-module of the total space of the dual bundle E ∨ → X, and (3) the quantum D-module of a submanifold Z ⊂ X cut out by a regular section of E. When E is the anticanonical line bundle K −1 X , we identify these twisted quantum D-modules with second structure connections with different parameters, which arise as Fourier-Laplace transforms of the quantum D-module of X. In this case, we show that the Duality Pairing is identified with Dubrovin's second metric. CONTENTS
Kim Wontae - One of the best experts on this subject based on the ideXlab platform.
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An existence result for nonhomogeneous quasilinear parabolic equations beyond the Duality Pairing
2019Co-Authors: Adimurthi Karthik, Byun Sun-sig, Kim WontaeAbstract:In this paper, we prove existence of \emph{very weak solutions} to nonhomogeneous quasilinear parabolic equations beyond the Duality Pairing. The main ingredients are a priori esitmates in suitable weighted spaces combined with the compactness argument developed in \cite{bulicek2018well}. In order to obtain the a priori estimates, we make use of the full Calder\'on-Zygmund machinery developed in the past few years and combine it with some sharp bounds for the subclass of Muckenhoupt weights considered in this paper.Comment: Updated references and fixed typos. arXiv admin note: text overlap with arXiv:1804.0435
