Academic literature on the topic 'MATHEMATICS / Differential Equations / Partial'

This Ph.D. thesis consists of five papers and an introduction to the main topics of the thesis. In Paper I we give an abstract criteria for existence of multiple solutions to nonlinear coupled equations involving magnetic Schrödinger operators. In paper II we establish existence of infinitely many solutions to the quasirelativistic Hartree-Fock equations for Coulomb systems along with properties of the solutions. In Paper III we establish existence of a ground state to the magnetic Hartree-Fock equations. In Paper IV we study the Choquard equation with general potentials (including quasirelativistic and magnetic versions of the equation) and establish existence of multiple solutions. In Paper V we prove that, under some assumptions on its nonmagnetic counterpart, a magnetic Schrödinger operator admits a representation with a positive Lagrange density and we derive consequences of this property.<br>I den tryckta boken har förlag felaktigt angivits som Acta Universitatis Upsaliensis.

Surface partial differential equations model several natural phenomena; for example in uid mechanics, cell biology and material science. The domain of the equations can often have complex and changing morphology. This implies analytic techniques are unavailable, hence numerical methods are required. The aim of this thesis is to design and analyse three methods for solving different problems with surface partial differential equations at their core. First, we define a new finite element method for numerically approximating solutions of partial differential equations in a bulk region coupled to surface partial differential equations posed on the boundary of this domain. The key idea is to take a polyhedral approximation of the bulk region consisting of a union of simplices, and to use piecewise polynomial boundary faces as an approximation of the surface and solve using isoparametric finite element spaces. We study this method in the context of a model elliptic problem. The main result in this chapter is an optimal order error estimate which is confirmed in numerical experiments. Second, we use the evolving surface finite element method to solve a Cahn- Hilliard equation on an evolving surface with prescribed velocity. We start by deriving the equation using a conservation law and appropriate transport formulae and provide the necessary functional analytic setting. The finite element method relies on evolving an initial triangulation by moving the nodes according to the prescribed velocity. We go on to show a rigorous well-posedness result for the continuous equations by showing convergence, along a subsequence, of the finite element scheme. We conclude the chapter by deriving error estimates and present various numerical examples. Finally, we stray from surface finite element method to consider new unfitted finite element methods for surface partial differential equations. The idea is to use a fixed bulk triangulation and approximate the surface using a discrete approximation of the distance function. We describe and analyse two methods using a sharp interface and narrow band approximation of the surface for a Poisson equation. Error estimates are described and numerical computations indicate very good convergence and stability properties.

The primary objective of this research is to investigate various optimization problems connected with partial differential equations (PDE). In chapter 2, we utilize the tool of tangent cones from convex analysis to prove the existence and uniqueness of a minimization problem. Since the admissible set considered in chapter 2 is a suitable convex set in $L^\infty(D)$, we can make use of tangent cones to derive the optimality condition for the problem. However, if we let the admissible set to be a rearrangement class generated by a general function (not a characteristic function), the method of tangent cones may not be applied. The central part of this research is Chapter 3, and it is conducted based on the foundation work mainly clarified by Geoffrey R. Burton with his collaborators near 90s, see [7, 8, 9, 10]. Usually, we consider a rearrangement class (a set comprising all rearrangements of a prescribed function) and then optimize some energy functional related to partial differential equations on this class or part of it. So, we call it rearrangement optimization problem (ROP). In recent years this area of research has become increasingly popular amongst mathematicians for several reasons. One reason is that many physical phenomena can be naturally formulated as ROPs. Another reason is that ROPs have natural links with other branches of mathematics such as geometry, free boundary problems, convex analysis, differential equations, and more. Lastly, such optimization problems also offer very challenging questions that are fascinating for researchers, see for example [2]. More specifically, Chapter 2 and Chapter 3 are prepared based on four papers [24, 40, 41, 42], mainly in collaboration with Behrouz Emamizadeh. Chapter 4 is inspired by [5]. In [5], the existence and uniqueness of solutions of various PDEs involving Radon measures are presented. In order to establish a connection between rearrangements and PDEs involving Radon measures, the author try to investigate a way to extend the notion of rearrangement of functions to rearrangement of Radon measures in Chapter 4.

This thesis consists of two parts, in both of which we consider approximations of rough stochastic PDEs and investigate convergence properties of the approximate solutions. In the first part we use the theory of (controlled) rough paths to define a solution for one-dimensional stochastic PDEs of Burgers type driven by an additive space-time white noise. We prove that natural numerical approximations of these equations converge to the solution of a corrected continuous equation and that their optimal convergence rate in the uniform topology (in probability) is arbitrarily close to 1/2 . In the second part of the thesis we develop a general framework for spatial discretisations of parabolic stochastic PDEs whose solutions are provided in the framework of the theory of regularity structures and which are functions in time. As an application, we show that the dynamical �43 model on the dyadic grid converges after renormalisation to its continuous counterpart. This result in particular implies that, as expected, the �43 measure is invariant for this equation and that the lifetime of its solutions is almost surely infinite for almost every initial condition.

In this thesis we present optimal and improved estimates for systems of critical elliptic PDE which arise as generalisations of natural geometric problems. We provide optimal regularity and compactness results for ‘Rivière’s equation’ for two dimensional domains via a new decay estimate, andwe exhibit examples to showthat the results are sharp. These results are presented in chapters 4 and 5. Such estimates generalise and improve known results in the classical setting. In chapter 6 we improve the known regularity for the higher dimensional theory introduced by Rivière-Struwe leading to better estimates for solutions in this case. Such estimates in particular lead to an easy proof for the regularity for stationary harmonic maps. We also present (in chapter 7) sharp results for a complex system of PDE, a consequence of which is a short proof of the full regularity for weakly harmonic maps from a Riemann surface into a closed Riemannian manifold.