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Operator Theory, Functional Analysis and Applications pp 235–253Cite as

Birkhäuser

Convergence Rates for Solutions of Inhomogeneous Ill-posed Problems in Banach Space with Sufficiently Smooth Data

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Part of the book series: Operator Theory: Advances and Applications ((OT,volume 282))

Abstract

We consider the inhomogeneous, ill-posed Cauchy problem

$$\displaystyle u'(t)=Au+h(t), \; 0<t<T, \quad u(0)=\varphi $$

where − A is the infinitesimal generator of a holomorphic semigroup of angle θ in Banach space. As in conventional regularization methods, certain auxiliary well-posed problems and their associated C 0 semigroups are applied in order to approximate a known solution u. A key property however, that the semigroups adhere to requisite growth orders, may fail depending on the value of the angle θ. Our results show that an approximation of u may be still be established in such situations as long as the data of the original problem is sufficiently smooth, i.e. in a small enough domain. Our results include well-known examples applied in the approach of quasi-reversibility as well as other types of approximations. The outcomes of the paper may be applied to partial differential equations in L p spaces, 1 < p < ∞ defined by strongly elliptic differential operators.

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Acknowledgements

The author would like to thank the editors for the proceedings of the IWOTA 2019 and the referee for their helpful suggestions.

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Authors and Affiliations

  1. Penn State Abington, Department of Mathematics, Abington, PA, USA

    Matthew A. Fury

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  1. Matthew A. Fury
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Correspondence to Matthew A. Fury .

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Editors and Affiliations

  1. Instituto Superior Técnico, Universidade de Lisboa, Lisboa, Portugal

    M. Amélia Bastos

  2. Departamento de Matemática, Universidade de Aveiro, Aveiro, Portugal

    Luís Castro

  3. Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa, Lisboa, Portugal

    Alexei Yu. Karlovich

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Fury, M.A. (2021). Convergence Rates for Solutions of Inhomogeneous Ill-posed Problems in Banach Space with Sufficiently Smooth Data. In: Bastos, M.A., Castro, L., Karlovich, A.Y. (eds) Operator Theory, Functional Analysis and Applications. Operator Theory: Advances and Applications, vol 282. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-51945-2_12

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