Multiplicity of periodic solutions in bistable equations

We study the number of periodic solutions in two first order non-autonomous differential equations both of which have been used to describe, among other things, the mean magnetization of an Ising magnet in the time-varying external magnetic field. When the strength of the external field is varied, the set of periodic solutions undergoes a bifurcation in both equations. We prove that despite profound similarities between the equations, the character of the bifurcation can be very different. This results in a different number of coexisting stable periodic solutions in the vicinity of the bifurcation. As a consequence, in one of the models, the Suzuki-Kubo equation, one can effect a discontinuous change in magnetization by adiabatically varying the strength of the magnetic field.Comment: Fixed typos; added and reordered figures. 18 pages, 6 figures. An animation of orbits is available at http://www.maths.strath.ac.uk/~aas02101/bistable

Homogenization Theory: Periodic and Beyond (online meeting)

The objective of the workshop has been to review the latest developments in homogenization theory for a large category of equations and settings arising in the modeling of solid, fluids, wave propagation, heterogeneous media, etc. The topics approached have covered periodic and nonperiodic deterministic homogenization, stochastic homogenization, regularity theory, derivation of wall laws and detailed study of boundary layers,..

프랙탈 영역 위에서의 비선형 작용소 및 비선형 포물형 편미분 방정식의 균질화

학위논문(박사) -- 서울대학교대학원 : 자연과학대학 수리과학부, 2021.8. 이기암.프랙탈 영역 위에서의 해석학은 해석적 접근과 확률론적 접근을 통해 다양하게 연구되고 있다. 본 학위논문에서는 프랙탈 영역에서 2차항을 포함하는 비선형 타원 방정식를 구성하고, 해석적 논증을 이용하여 해의 정칙성을 구하고자 한다. 프랙탈 영역에서는 기존의 편미분 이론을 사용할 수 없기 때문에, 우리의 접근 방식은 그래프 근사 논증을 이용하여 디리클레 형식을 구성하는 것에 기반을 두고 있다. 가장 중점적인 개념은 프랙탈 영역의 특수한 기하학적 특성을 사용하여 적절한 차단 함수와 가중치 부등식을 찾는 것이다. 본 학위논문의 또 다른 주제는 완전 비선형 포물형 방정식에 대한 균질화 이론이다. 특히, 우리는 진동 변수들의 척도가 기존과 다른 경우에 대해서 다룬다. 흥미로운 점은 시공간 빠른 변수의 척도가 일치하지 않기 때문에 균질화가 시간과 공간에 대해 개별적으로 발생한다는 점이다. 또한 이 현상은 기존과 다른 수렴속도를 야기한다.The analysis of fractals has been studied extensively in both analysis and probability approaches. In this thesis, we construct the non-linear elliptic equation involving second order terms on fractal spaces, and our main object is to exhibit the regularity of their solutions by using an analytic argument. Since a calculus on fractals is not available, our approach is based on the graph approximation argument to construct Dirichlet forms. The central concept is in finding suitable cut-off functions and weighted inequalities, which can be obtained by using the special geometric properties of the fractal domain. Another topic in this thesis is the homogenization theory for fully non-linear parabolic equations. In particular, we treat the case with different scales of the oscillating variables. The interesting point is that the homogenization occurs separately for time and space due to a mismatch in the scale of time and space fast variables. In addition, this phenomenon causes different order of convergence rates.1 Introduction 1 1.1 Part I : Non-linear operators on the fractal domains 1 1.2 Part II : Homogenization for fully non-linear parabolic equations 3 2 Preliminaries 7 2.1 Part I : Non-linear operators on the fractal domains 7 2.1.1 Sierpinski gasket 7 2.1.2 Dirichlet forms and harmonic functions 9 2.2 Part II : Homogenization for fully non-linear parabolic equations 15 2.2.1 Cell problem 15 2.2.2 Effective operators and e effective limits 20 3 Non-linear operators of divergence form on the Sierpinski gasket 26 3.1 Introduction 26 3.1.1 Main results 27 3.1.2 Main strategies 28 3.1.3 Outline 30 3.2 L-harmonic functions 30 3.3 Weighted inequalities 37 3.3.1 Barriers 38 3.3.2 Weighted inequalities 40 3.4 Harnack inequality 55 3.4.1 Caccioppoli type inequality and local boundedness 56 3.4.2 Harnack inequality 64 4 Homogenization of fully non-linear parabolic equations with different oscillations in space and time 73 4.1 Introduction 73 4.1.1 Main results 75 4.1.2 Heuristics discussion and main strategies 78 4.1.3 Outline 84 4.2 basic homogenization process 84 4.3 Homogenization when k \in (0,2) 86 4.3.1 The effective operator and the effective limit 86 4.3.2 Rate of convergence for the homogenization 90 4.4 Homogenization when k \in (2,\infty) 104 4.4.1 The effective operator and the effective limit 104 4.4.2 Rate of convergence for the homogenization 108 5 Higher order convergence rate for the homogenization of soft inclusions with non-divergence structure 121 5.1 Introduction 121 5.1.1 Main results 124 5.1.2 Heuristics discussion and main strategies 127 5.1.3 Outline 128 5.2 Homogenization and correctors 128 5.2.1 Basic homogenization process and regularity of solutions 128 5.2.2 Asymptotic expansions and correctors 139 5.2.3 Higher order interior correctors 145 5.3 Higher order convergence rate 153 Bibliography 159 Abstract (in Korean) 166 Acknowledgement (in Korean) 167박

Sticky particle dynamics with interactions

We consider compressible pressureless fluid flows in Lagrangian coordinates in one space dimension. We assume that the fluid self-interacts through a force field generated by the fluid itself. We explain how this flow can be described by a differential inclusion on the space of transport maps, in particular when a sticky particle dynamics is assumed. We study a discrete particle approximation and we prove global existence and stability results for solutions of this system. In the particular case of the Euler-Poisson system in the attractive regime our approach yields an explicit representation formula for the solutions

EQUADIFF 15

Equadiff 15 – Conference on Differential Equations and Their Applications – is an international conference in the world famous series Equadiff running since 70 years ago. This booklet contains conference materials related with the 15th Equadiff conference in the Czech and Slovak series, which was held in Brno in July 2022. It includes also a brief history of the East and West branches of Equadiff, abstracts of the plenary and invited talks, a detailed program of the conference, the list of participants, and portraits of four Czech and Slovak outstanding mathematicians

Excitation of surface plasmon-polaritons in metal films with double periodic modulation: anomalous optical effects

We perform a thorough theoretical analysis of resonance effects when an arbitrarily polarized plane monochromatic wave is incident onto a double periodically modulated metal film sandwiched by two different transparent media. The proposed theory offers a generalization of the theory that had been build in our recent papers for the simplest case of one-dimensional structures to two-dimensional ones. A special emphasis is placed on the films with the modulation caused by cylindrical inclusions, hence, the results obtained are applicable to the films used in the experiments. We discuss a spectral composition of modulated films and highlight the principal role of ``resonance'' and ``coupling'' modulation harmonics. All the originating multiple resonances are examined in detail. The transformation coefficients corresponding to different diffraction orders are investigated in the vicinity of each resonance. We make a comparison between our theory and recent experiments concerning enhanced light transmittance and show the ways of increasing the efficiency of these phenomena. In the appendix we demonstrate a close analogy between ELT effect and peculiarities of a forced motion of two coupled classical oscillators.Comment: 24 pages, 17 figure