scholarly journals Resolvent Estimates for the Lamé Operator and Failure of Carleman Estimates

2021 ◽  
Vol 27 (3) ◽  
Author(s):  
Yehyun Kwon ◽  
Sanghyuk Lee ◽  
Ihyeok Seo
Keyword(s):  
Carleman Estimates ◽  
Resolvent Estimates

Stability in 𝐿𝑞-norm for inverse source parabolic problems

2020 ◽  
Vol 28 (6) ◽  
pp. 797-814
Author(s):  
Elena-Alexandra Melnig
Keyword(s):  
Parabolic Equations ◽  
Main Tool ◽  
Parabolic Systems ◽  
Carleman Estimates ◽  
Parabolic Problems ◽  
Lebesgue Spaces ◽  
First Order ◽  
Lipschitz Estimates ◽  
Systems Of Parabolic Equations

AbstractWe consider systems of parabolic equations coupled in zero and first order terms. We establish Lipschitz estimates in {L^{q}}-norms, {2\leq q\leq\infty}, for the source in terms of the solution in a subdomain. The main tool is a family of appropriate Carleman estimates with general weights, in Lebesgue spaces, for nonhomogeneous parabolic systems.

Inverse problem for a degenerate/singular parabolic system with Neumann boundary conditions

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Mohammed Alaoui ◽  
Abdelkarim Hajjaj ◽  
Lahcen Maniar ◽  
Jawad Salhi
Keyword(s):  
Boundary Conditions ◽  
Parabolic System ◽  
Source Term ◽  
Neumann Boundary ◽  
Stability Estimate ◽  
Carleman Estimates ◽  
Well Posedness ◽  
Degenerate Diffusion ◽  
Neumann Type ◽  
Lower Order Terms

AbstractIn this paper, we study an inverse source problem for a degenerate and singular parabolic system where the boundary conditions are of Neumann type. We consider a problem with degenerate diffusion coefficients and singular lower-order terms, both vanishing at an interior point of the space domain. In particular, we address the question of well-posedness of the problem, and then we prove a stability estimate of Lipschitz type in determining the source term by data of only one component. Our method is based on Carleman estimates, cut-off procedures and a reflection technique.

scholarly journals Eigenvalue bounds for non-selfadjoint Dirac operators

2021 ◽  
Author(s):  
Piero D’Ancona ◽  
Luca Fanelli ◽  
Nico Michele Schiavone
Keyword(s):  
Dirac Operator ◽  
Discrete Spectrum ◽  
Complex Plane ◽  
Dirac Operators ◽  
Resolvent Estimates ◽  
Eigenvalue Bounds ◽  
Mixed Norms ◽  
Massless Case ◽  
Abstract Version ◽  
Embedded Eigenvalues

AbstractWe prove that the eigenvalues of the n-dimensional massive Dirac operator $${\mathscr {D}}_0 + V$$ D 0 + V , $$n\ge 2$$ n ≥ 2 , perturbed by a potential V, possibly non-Hermitian, are contained in the union of two disjoint disks of the complex plane, provided V is sufficiently small with respect to the mixed norms $$L^1_{x_j} L^\infty _{{\widehat{x}}_j}$$ L x j 1 L x ^ j ∞ , for $$j\in \{1,\dots ,n\}$$ j ∈ { 1 , ⋯ , n } . In the massless case, we prove instead that the discrete spectrum is empty under the same smallness assumption on V, and in particular the spectrum coincides with the spectrum of the unperturbed operator: $$\sigma ({\mathscr {D}}_0+V)=\sigma ({\mathscr {D}}_0)={\mathbb {R}}$$ σ ( D 0 + V ) = σ ( D 0 ) = R . The main tools used are an abstract version of the Birman–Schwinger principle, which allows in particular to control embedded eigenvalues, and suitable resolvent estimates for the Schrödinger operator.

Carleman estimates for parabolic equations and applications

2009 ◽  
Vol 25 (12) ◽  
pp. 123013 ◽  
Author(s):  
Masahiro Yamamoto
Keyword(s):  
Parabolic Equations ◽  
Carleman Estimates

scholarly journals Global Carleman Estimates for Waves and Applications

2013 ◽  
Vol 38 (5) ◽  
pp. 823-859 ◽  
Author(s):  
Lucie Baudouin ◽  
Maya de Buhan ◽  
Sylvain Ervedoza
Keyword(s):  
Carleman Estimates
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