Citation
Lopes, Louis Aloysius (1964) Operator Differential Equations in Hilbert Space. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/1N8NY957. https://resolver.caltech.edu/CaltechETD:etd10182002082821
Abstract
NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document. In this paper the theory of dissipative linear operators in Hilbert space developed by R. S. Phillips has been applied in the study of the Cauchy problem [...](t) + A(t)x(t) = f(t), x(o) = x[subscript o] where A(t), t [epsilon] [o,[tau]], is a family of unbounded linear operators with a common dense domain D in a Hilbert space H, f [epsilon] [...], the Hilbert space of measurable functions on [o, [tau]] with values in H which have square integrable norm, and x[subscript o] [epsilon] H. It is assumed that for each t [epsilon] [o,[tau]] A(t) is maximal dissipative, satisfying for each x [epsilon] D, Re (A(t)x,x) [greater than or equal to] [alpha] [...], [alpha] > o, and A(t)x is strongly continuous and has a bounded measurable strong derivative on J. Let A[subscript o] be any maximal dissipative linear operator with domain D satisfying Re (A[subscript o]x,x) [greater than or equal to] [alpha] [...] for all x [epsilon] D. Then B(t) = A(t)A[subscript o][superscript 1] is a onetoone continuous linear transformation of H onto itself. It is assumed that B[superscript 1](t) is bounded on [o, [tau]]. Under these conditions it is shown that, first, there exists a weak solution to the Cauchy problem, and, second, that the weak solution is a unique strong solution which is the limit of a sequence of classical solutions. The theory is applied to a timedependent hyperbolic system of partial differential equations.
Item Type:  Thesis (Dissertation (Ph.D.)) 

Subject Keywords:  (Mathematics) 
Degree Grantor:  California Institute of Technology 
Division:  Physics, Mathematics and Astronomy 
Major Option:  Mathematics 
Thesis Availability:  Public (worldwide access) 
Research Advisor(s): 

Thesis Committee: 

Defense Date:  19 May 1964 
Record Number:  CaltechETD:etd10182002082821 
Persistent URL:  https://resolver.caltech.edu/CaltechETD:etd10182002082821 
DOI:  10.7907/1N8NY957 
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided. 
ID Code:  4148 
Collection:  CaltechTHESIS 
Deposited By:  Imported from ETDdb 
Deposited On:  21 Oct 2002 
Last Modified:  19 Jan 2024 22:56 
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