Mathematical Sciences Dissertations
https://hdl.handle.net/20.500.12202/5754
2020-08-04T17:02:43ZQualitative Properties for Positive Solutions of Nonlocal Equations.
https://hdl.handle.net/20.500.12202/5805
Qualitative Properties for Positive Solutions of Nonlocal Equations.
Hu, Yunyun
This thesis is devoted to the study of properties for nonnegative solutions to nonlocal
problems and integral equations. The main tools we use are the method of moving planes
and the method of moving spheres._____________
First, we focus on the nonlocal problems involving fractional p-Laplacian (p 2)
in unbounded domains. Without assuming any asymptotic behavior of positive solutions
near in nity, we develop narrow region principles in unbounded domains, then using the
method of moving planes, we establish the monotonicity of positive solutions.__________
Second, we study the symmetry of positive solutions for nonlinear equations involving
fractional Laplacian. In bounded domain, we prove that all positive solutions of
fractional equations with Hardy Leray potential are radically symmetric about the origin.
Then we consider a nonlocal problem in unbounded cylinders. By using the method of
moving planes, we establish the symmetry and monotonicity of positive solutions. Furthermore,
we obtain the nonexistence of nonnegative solutions for nonlocal problems in
the whole space Rn.___________
Third, we establish a strong maximum principle and a Hopf type lemma for antisymmetric
solutions of fractional parabolic equations in unbounded domains. These will
become most commonly used basic techniques in the study of monotonicity and symmetry
of solutions.________
Finally, we consider general integral systems on a half space and integral equations
in bounded domains. Under natural integrability conditions, we obtain a classi cation
of positive solutions for an integral system on half space by using a slight variant of
the method of moving spheres. Here we removed the global integrability hypothesis on
positive solutions by introducing some new ideas. In addition, we study the symmetry
and monotonicity of positive solutions to over-determined problems and partially overdetermined
problems. The main technique we use is the method of moving planes in an
integral form.
Doctoral Dissertation, Ph.D. --- Opt-Out. For access, please contact: yair@yu.edu
2020-05-28T00:00:00ZClassical mechanics over fields of characteristic p greater than 0
https://hdl.handle.net/20.500.12202/3200
Classical mechanics over fields of characteristic p greater than 0
Merewether, James William
A formulation of the algebraic structures of a classical mechanics over fields of characteristic p {dollar}>{dollar} 0 is presented. It is shown how a definition of an abstract mechanics, which is a composition class of two-product algebras, allows for realizations over fields of characteristic p {dollar}>{dollar} 0. A broad class of such realizations of simple two-product algebras is derived and their structure examined. They are shown to be simple nodal noncommutative Jordan algebras, defined by a non-degenerate skew-symmetric bilinear form, over the field of intergers modulo p (p prime). Equations for determination of the automorphism group of the algebras are deduced. The affine restriction of the canonical group is given explicitly. Finally, a characteristic p {dollar}>{dollar} 0 analog of the Galilei group is given and its canonical realizations are produced.
1986-01-01T00:00:00ZTHE SCATTERING THEORY OF THE KLEIN-GORDON EQUATION IN TWO HILBERT SPACES WITH GENERAL AND OSCILLATING POTENTIALS
https://hdl.handle.net/20.500.12202/2983
THE SCATTERING THEORY OF THE KLEIN-GORDON EQUATION IN TWO HILBERT SPACES WITH GENERAL AND OSCILLATING POTENTIALS
GELMAN, ALEXANDER
This dissertation considers three problems associated with the Klein-Gordon Equation: (a) The conditions for the operator to be self-adjoint; (b) The existence of the wave operator; and (c) The completeness of the wave operator. These problems are considered for the operator with general and oscillating potentials.;For problem (a) the work is based on the theory of forms extensions originated by K. Friederichs; and for problems (b) and (c), the abstract theory of scattering which originated in the work of Kato and Birman. The particular result which we use for problems (a) and (b) is the recent theorem proven by M. Schechter, in which he was able to relax requirements on J (no requirement for the bijectivity of J, and no reference to R(z), for example).;Application of the methods described above to the Klein-Gordon operator allowed us to solve the three problems above for an unbounded operator J and also for the oscillating potential.
1984-01-01T00:00:00ZTHE SPECTRA OF THE SCHROEDINGER OPERATOR
https://hdl.handle.net/20.500.12202/2976
THE SPECTRA OF THE SCHROEDINGER OPERATOR
BRENNER, TERENCE
We look at the Schrodinger operator H=-(DELTA)+q(x) where (DELTA) is the Laplacian and q(x)(epsilon)R('n). We give sufficient conditions for the spectrum of H to contain the interval of the form {lcub}a,(INFIN)) and sufficient conditions for the essential spectrum of H to contain the interval of the form {lcub}b,(INFIN)). Our estimates for the lower bounds of a and b are positive numbers. We allow q(x) to be negative in some region. Our results are in R('2) and in R('n).
1984-01-01T00:00:00Z