Carleman estimates with sharp weights and boundary observability for wave operators with critically singular potentials

AbstractWe obtain approximation bounds for products of quasimodes for the Laplace–Beltrami operator on compact Riemannian manifolds of all dimensions without boundary. We approximate the products of quasimodes uv by a low-degree vector space {B_{n}}, and we prove that the size of the space {\dim(B_{n})} is small. In this paper, we first study bilinear quasimode estimates of all dimensions {d=2,3}, {d=4,5} and {d\geq 6}, respectively, to make the highest frequency disappear from the right-hand side. Furthermore, the result of the case {\lambda=\mu} of bilinear quasimode estimates improves {L^{4}} quasimodes estimates of Sogge and Zelditch in [C. D. Sogge and S. Zelditch, A note on L^{p}-norms of quasi-modes, Some Topics in Harmonic Analysis and Applications, Adv. Lect. Math. (ALM) 34, International Press, Somerville 2016, 385–397] when {d\geq 8}. And on this basis, we give approximation bounds in {H^{-1}}-norm. We also prove approximation bounds for the products of quasimodes in {L^{2}}-norm using the results of {L^{p}}-estimates for quasimodes in [M. Blair, Y. Sire and C. D. Sogge, Quasimode, eigenfunction and spectral projection bounds for Schrodinger operators on manifolds with critically singular potentials, preprint 2019, https://arxiv.org/abs/1904.09665]. We extend the results of Lu and Steinerberger in [J. F. Lu and S. Steinerberger, On pointwise products of elliptic eigenfunctions, preprint 2018, https://arxiv.org/abs/1810.01024v2] to quasimodes.

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Correction to: Optimal Control of a Phase Field System Modelling Tumor Growth with Chemotaxis and Singular Potentials

Applied Mathematics & Optimization ◽

10.1007/s00245-021-09771-x ◽

2021 ◽

Author(s):

Pierluigi Colli ◽

Andrea Signori ◽

Jürgen Sprekels

Keyword(s):

Optimal Control ◽

Tumor Growth ◽

Phase Field ◽

System Modelling ◽

Singular Potentials ◽

Field System ◽

Phase Field System

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Stability in 𝐿𝑞-norm for inverse source parabolic problems

Journal of Inverse and Ill-Posed Problems ◽

10.1515/jiip-2019-0081 ◽

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.

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Resolvent Estimates for the Lamé Operator and Failure of Carleman Estimates

Journal of Fourier Analysis and Applications ◽

10.1007/s00041-021-09859-6 ◽

2021 ◽

Vol 27 (3) ◽

Author(s):

Yehyun Kwon ◽

Sanghyuk Lee ◽

Ihyeok Seo

Keyword(s):

Carleman Estimates ◽

Resolvent Estimates

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

Journal of Inverse and Ill-Posed Problems ◽

10.1515/jiip-2018-0034 ◽

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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