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Study Of The Equilibria Of Parabolic Differential Equations With Interfaces Intersecting The Boundary, Michal Kowalczyk
Study Of The Equilibria Of Parabolic Differential Equations With Interfaces Intersecting The Boundary, Michal Kowalczyk
Doctoral Dissertations
Existence of steady state solutions for the Allen-Cahn and Cahn-Hilliard equations in two dimensional domains is discussed. We are in particular interested in establishing existence of single layered equilibria with the property that their transition layer intersects the boundary. In the case of the Allen-Cahn equation we consider bone-like domains and seek solutions intersecting the flat part of the boundary. We establish conditions for the domain which ensure existence of such equilibria. Their stability is also analyzed. For the Cahn-Hilliard equations we show that there exist equilibria near every point of a local maximum of the curvature of the boundary.
On LP Solutions Of Second Order Linear Differential Equations, James C. Smith
On LP Solutions Of Second Order Linear Differential Equations, James C. Smith
Doctoral Dissertations
In this dissertation we study the Lp solutions of second order linear differential equations. The question as to when the equation -(qo(x)y'(x))' + q1(x)y(x) = f(x), α ≤ x < ∞, admits Lp solutions y(x) for arbitrary f(x) in Lp is investigated. We show the condition Re(q1(x)) ≥ 1 or the conditions Re(q1(x)) ≥ 0 and Im(q1(x)) ≥ 1 are sufficient for a Lp solution y(x) to exist.
Functions that bound a solution of the homogeneous equation …