scholarly journals Cohomology with bounds and Carleman estimates for the∂¯-operator on Stein manifolds

2002 ◽  
Vol 32 (6) ◽  
pp. 383-386
Author(s):  
Patrick W. Darko
Keyword(s):  
Carleman Estimates ◽  
Stein Manifolds ◽  
Cauchy Riemann

Cohomology with bounds are used to globalize a result of Hörmander obtaining Carleman estimates for the Cauchy-Riemann operator on Stein manifolds.

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.

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