TY - THES
AU - Richardson, David
PY - 1978
TI - Variational problems with thin obstacles
KW - Thesis/Dissertation
LA - eng
M3 - Text
AB - In this thesis the solution to the variational problem of Signorini is studied, namely:
(i) Δv = 0 in Ω; (ii) v ≥ ѱ on əΩ; (iii) əv/əѵ ≤ g on əΩ; (iv) (v- ѱ) (əv/əѵ – g) = 0 on əΩ
where Ω is a domain in R[sup n], and v is the unit inner normal vector to əΩ.
In the case n = 2 a regularity theorem is proved.
It is shown that if ѱ Є C[sup 1,α] (əΩ), g Є Lip α(əΩ) then v Є C[sup 1,α] (əΩ) if α < 1/2 . An example is given to shown that this result is optimal. The method of proof relies on techniques of complex analysis and therefore does not extend to higher dimensions.
For n > 2 the case where Ω, is unbounded, or equivalently, where ѱ is unbounded in a neighbourhood of some point of əΩ is considered. This is a situation where known existence theorems do not apply. Some sufficient conditions for the pair (ѱ,g) are derived that will ensure the existence of a solution in this case, thereby extending some results obtained by A. Beurling and P. Malliavin in the two dimensional case. The proof involves a variational problem in a Hilbert space analogous to the one considered by Beurling and Malliavin, and some pointwise estimates of Riesz transforms.
N2 - In this thesis the solution to the variational problem of Signorini is studied, namely:
(i) Δv = 0 in Ω; (ii) v ≥ ѱ on əΩ; (iii) əv/əѵ ≤ g on əΩ; (iv) (v- ѱ) (əv/əѵ – g) = 0 on əΩ
where Ω is a domain in R[sup n], and v is the unit inner normal vector to əΩ.
In the case n = 2 a regularity theorem is proved.
It is shown that if ѱ Є C[sup 1,α] (əΩ), g Є Lip α(əΩ) then v Є C[sup 1,α] (əΩ) if α < 1/2 . An example is given to shown that this result is optimal. The method of proof relies on techniques of complex analysis and therefore does not extend to higher dimensions.
For n > 2 the case where Ω, is unbounded, or equivalently, where ѱ is unbounded in a neighbourhood of some point of əΩ is considered. This is a situation where known existence theorems do not apply. Some sufficient conditions for the pair (ѱ,g) are derived that will ensure the existence of a solution in this case, thereby extending some results obtained by A. Beurling and P. Malliavin in the two dimensional case. The proof involves a variational problem in a Hilbert space analogous to the one considered by Beurling and Malliavin, and some pointwise estimates of Riesz transforms.
UR - https://open.library.ubc.ca/collections/831/items/1.0080165
ER - End of Reference