Leray-Schauder degree theory and partial differential equations under nonlinear boundary conditions

Cranny, Timothy R. (Timothy Roy), 1968- (1992). Leray-Schauder degree theory and partial differential equations under nonlinear boundary conditions PhD Thesis, School of Physical Sciences, University of Queensland. doi:10.14264/uql.2016.434

       
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Author Cranny, Timothy R. (Timothy Roy), 1968-
Thesis Title Leray-Schauder degree theory and partial differential equations under nonlinear boundary conditions
School, Centre or Institute School of Physical Sciences
Institution University of Queensland
DOI 10.14264/uql.2016.434
Publication date 1992
Thesis type PhD Thesis
Supervisor H.B.Thompson
Total pages 189
Language eng
Subjects 01 Mathematical Sciences
Formatted abstract
We consider the quasilinear elliptic partial differential equation 

Qu(x) = aij(x, u(x), Du(x))Diju(x) + a(x, u(x), Du(x)) = 0 in Ω  (0.1) 

with the imposed nonlinear boundary condition 

Gu(x) = g(x, u(x), Du(x)) = 0 on ∂Ω.                                       (0.2) 

A homotopy argument for the Leray-Schauder degree is presented which reduces the question of existence of solutions to the above problem to that of establishing suitable a priori bounds on solutions of an approximating problem. 

The above problem is re-written as

aij(x,u, Du)DijU + a(x,u, Du) = 0 in Ω
                                                                                               (0.3)
                                     u(x) = ω(x) on ∂Ω.


where ω(.) is a function which satisfies the requirement that 

g(x,ω(x), Du(x)) = 0 on ∂Ω.                                                    (0.4)

The gradient term in the boundary condition is replaced with an approximating function of greater smoothness, allowing the application of Leray-Schauder degree theory. We present simple assumptions which ensure the existence of both an upper solution and a lower solution of the differential operator Q, and impose simple conditions upon the boundary operator g(x,z,p) which allow us to use a homotopy argument to reduce the problem to two 'disconnected' halves, one of which is a simple Dirichlet problem in u with no reference to the function ω, the other an equation in ω with no reference to u or its derivatives. A reduction argument and reasonable assumptions upon g(x,z,p) then imply the existence of solutions to the approximating problems. 

It is then shown that an a priori bound on the solutions to the approximating equations in a Hölder space of the form C1,α(Ω) where α > 0 suffices to ensure the convergence of a subsequence of these functions to a classical solution of the original problem. 

The result described here differ from the method of continuity in that the boundary ∂Ω, is assumed to be only C1,α, and the monotonicity of the differential operator with respect to u is significantly relaxed. The techniques described also apply in situations where degree theory cannot be applied directly.
Keyword Differential equations, Elliptic
Differential equations, Nonlinear
Boundary value problems
Additional Notes Spine title: Degree theory and elliptic partial differential equations.

Document type: Thesis
Collection: UQ Theses (RHD) - UQ staff and students only
 
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