Felix Finster
We formulate a variational principle for a collection of projectors in an indefinite inner product space. The existence of minimizers is proved in various situations.
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We prove the existence of minimizers in the class of negative definite measures on compact subsets of momentum space in the homogeneous setting under several side conditions (constraints). The method is to employ Prohorov's theorem. Given a minimizing sequence of negative definite measures, we show that, under suitable side conditions, a unitarily equivalent subsequence thereof is bounded. By restricting attention to compact subsets, from Prohorov's theorem we deduce the existence of minimizers in the class of negative definite measures.
Considering second variations about a given minimizer of a causal variational principle, we derive positive functionals in space-time. It is shown that the strict positivity of these functionals ensures that the minimizer is nonlinearly stable within the class of compactly supported variations with local fragmentation. As applications, we endow the space of jets in space-time with Hilbert space structures and derive a positive surface layer integral on solutions of the linearized field equations.
The electromagnetic two-body problem has \emph{neutral differential delay} equations of motion that, for generic boundary data, can have solutions with \emph{discontinuous} derivatives. If one wants to use these neutral differential delay equations with \emph{arbitrary} boundary data, solutions with discontinuous derivatives must be expected and allowed. Surprisingly, Wheeler-Feynman electrodynamics has a boundary value variational method for which minimizer trajectories with discontinuous derivatives are also expected, as we show here. The variational method defines continuous trajectories wi…
We derive the Euler-Lagrange equations for minimizers of causal variational principles in the non-compact setting with constraints, possibly prescribing symmetries. Considering first variations, we show that the minimizing measure is supported on the intersection of a hyperplane with a level set of a function which is homogeneous of degree two. Moreover, we perform second variations to obtain that the compact operator representing the quadratic part of the action is positive semi-definite. The key ingredient for the proof is a subtle adaptation of the Lagrange multiplier method to variational …
We provide a method for constructing (possibly non-trivial) measures on non-locally compact Polish subspaces of infinite-dimensional separable Banach spaces which, under suitable assumptions, are minimizers of causal variational principles in the non-locally compact setting. Moreover, for non-trivial minimizers the corresponding Euler-Lagrange equations are derived. The method is to exhaust the underlying Banach space by finite-dimensional subspaces and to prove existence of minimizers of the causal variational principle restricted to these finite-dimensional subsets of the Polish space under …
We prove the existence of minimizers of causal variational principles on second countable, locally compact Hausdorff spaces. Moreover, the corresponding Euler-Lagrange equations are derived. The method is to first prove the existence of minimizers of the causal variational principle restricted to compact subsets for a lower semi-continuous Lagrangian. Exhausting the underlying topological space by compact subsets and rescaling the corresponding minimizers, we obtain a sequence which converges vaguely to a regular Borel measure of possibly infinite total volume. It is shown that, for continuous…
Periodic waves of the modified Korteweg-de Vries (mKdV) equation are identified in the context of a new variational problem with two constraints. The advantage of this variational problem is that its non-degenerate local minimizers are stable in the time evolution of the mKdV equation, whereas the saddle points are unstable. We explore the analytical representation of periodic waves given by Jacobi elliptic functions and compute numerically critical points of the constrained variational problem. A broken pitchfork bifurcation of three smooth solution families is found. Two families represent (…
The support of minimizing measures of the causal variational principle on the sphere is analyzed. It is proven that in the case $τ>\sqrt{3}$, the support of every minimizing measure is contained in a finite number of real analytic curves which intersect at a finite number of points. In the case $τ>\sqrt{6}$, the support is proven to have Hausdorff dimension at most $6/7$.
A class of causal variational principles on a compact manifold is introduced and analyzed both numerically and analytically. It is proved under general assumptions that the support of a minimizing measure is either completely timelike, or it is singular in the sense that its interior is empty. In the examples of the circle, the sphere and certain flag manifolds, the general results are supplemented by a more detailed and explicit analysis of the minimizers. On the sphere, we get a connection to packing problems and the Tammes distribution. Moreover, the minimal action is estimated from above a…
Variational interacting particle systems and Vlasov equations
本文研究了相互作用粒子系统的优化问题，证明了临界点满足Vlasov方程，并将极小解与Hamilton-Jacobi-Bellman方程联系起来。
Minimization of the discrete interaction energy with smooth potentials
本文研究平坦环面上正且快速衰减 Fourier 系数函数的离散成对相互作用能，证明一维最小化配置为等距点集、二维为三角晶格。
Existence and uniqueness of minimizers for axisymmetric nematic films
本文研究了轴对称向列膜的变分模型，证明了最小化器的存在性和唯一性，并通过数值模拟验证了结果。
The Bogoliubov-Bose-Hubbard model: existence of minimizers and absence of quantum phase transition
该研究基于博利尤布理论分析Bose-Hubbard模型，证明了极小值的存在性并展示热驱动相变，同时指出该模型不存在量子相变。
In 1979, G. Parisi predicted a variational formula for the thermodynamic limit of the free energy in the Sherrington-Kirkpatrick model and described the role played by its minimizer. This formula was verified in the seminal work of Talagrand and later generalized to the mixed p-spin models by Panchenko. In this paper, we prove that the minimizer in Parisi's formula is unique at any temperature and external field by establishing the strict convexity of the Parisi functional.
By variational methods, we prove the inequality: $$ \int_{\mathbb{R}} u''{}^2 dx-\int_{\mathbb{R}} u'' u^2 dx\geq I \int_{\mathbb{R}} u^4 dx\quad \forall u\in L^4({\mathbb{R}}) {such that} u''\in L^2({\mathbb{R}}) $$ for some constant $I\in (-9/64,-1/4)$. This inequality is connected to Lieb-Thirring type problems and has interesting scaling properties. The best constant is achieved by sign changing minimizers of a problem on periodic functions, but does not depend on the period. Moreover, we completely characterize the minimizers of the periodic problem.
We introduce a class of variational principles on measure spaces which are causal in the sense that they generate a relation on pairs of points, giving rise to a distinction between spacelike and timelike separation. General existence results are proved. It is shown in examples that minimizers need not be unique. Counter examples to compactness are discussed. The existence results are applied to variational principles formulated in indefinite inner product spaces.
We study the non-autonomous variational problem: \begin{equation*} \inf_{(ϕ,θ)} \bigg\{\int_0^1 \bigg(\frac{k}{2}ϕ'^2 + \frac{(ϕ-θ)^2}{2}-V(x,θ)\bigg)\text{d}x\bigg\} \end{equation*} where $k>0$, $V$ is a bounded continuous function, $(ϕ,θ)\in H^1([0,1])\times L^2([0,1])$ and $ϕ(0)=0$ in the sense of traces. The peculiarity of the problem is its setting in the product of spaces of different regularity order. Problems with this form arise in elastostatics, when studying the equilibria of a nonlinear Timoshenko beam under distributed load, and in classical dynamics of coupled particles in time-d…
Discrete minimizers of the interaction energy in collective behavior: a brief numerical and analytic review
本文综述了集体行为模型中N粒子相互作用能量极小化子的数值计算方法，并用GMIN软件展示其在低粒子数下对包括Lennard-Jones及较不奇异势的有效性，同时数值揭示了二维极小化子的晶格结构、Wulff形状和大N连续极限。
Existence and large-field asymptotics of Hopfions in a continuum model of frustrated magnets
该研究针对阻挫磁体连续统模型，证明了无穷多非平凡同伦类能量极小值、强外场下单位电荷霍普夫子的存在性，还分析了大场极限并证明重标极小值的强收敛性。
Periodic waves in the fractional Korteweg-de Vries equation have been previously characterized as constrained minimizers of energy subject to fixed momentum and mass. Here we characterize these periodic waves as constrained minimizers of the quadratic form of energy subject to fixed cubic part of energy and the zero mean. This new variational characterization allows us to unfold the existence region of travelling periodic waves and to give a sharp criterion for spectral stability of periodic waves with respect to perturbations of the same period. The sharp stability criterion is given by the m…
Equilibrium shapes of two-dimensional charged, perfectly conducting liquid drops are governed by a geometric variational problem that involves a perimeter term modeling line tension and a capacitary term modeling Coulombic repulsion. Here we give a complete explicit solution to this variational problem. Namely, we show that at fixed total charge a ball of a particular radius is the unique global minimizer among all sufficiently regular sets in the plane. For sets whose area is also fixed, we show that balls are the only minimizers if the charge is less than or equal to a critical charge, while…
A variational model of charged drops in dielectrically matched binary fluids: the effect of charge discreteness
本文通过引入基本电荷离散性修正经典 Rayleigh 带电液滴模型的不适定性，证明广义极小化子的存在性，并给出显著低于 Rayleigh 模型的电荷数量阈值及两电荷宏观大液滴的显式解。
Integrability in the weak noise theory
本研究证明随机热方程Freidlin–Wentzell大偏差变分问题的极小元满足可积的虚时间非线性薛定谔方程组，验证了已有物理公式的正确性，并推导了带delta初值情形的单点速率函数显式表达式及最概然形状估计。
We analyze an elastic surface energy which was recently introduced by G. Napoli and L.Vergori to model thin films of nematic liquid crystals. We show how a novel approach that takes into account also the extrinsic properties of the surfaces coated by the liquid crystal leads to considerable differences with respect to the classical intrinsic energy. Our results concern three connected aspects: i) using methods of the calculus of variations, we establish a relation between the existence of minimizers and the topology of the surface; ii) we prove, by a Ginzburg-Landau approximation, the well-pos…
Gradient Existence and Energy Finiteness of Local Minimizers in the Wasserstein $L^\infty$ Topology for Binary-Star Systems
本文在Wasserstein L^∞拓扑下研究二星系统局部极小解的梯度存在性和能量有限性，扩展了McCann的结果，探讨了局部极小解的三个关键性质。
We consider the variational problem associated with the Freidlin--Wentzell Large Deviation Principle of the Stochastic Heat Equation (SHE). The logarithm of the minimizer of the variational problem gives the most probable shape of the solution of the Kardar--Parisi--Zhang equation conditioned on achieving certain unlikely values. Taking the SHE with the delta initial condition and conditioning the value of its solution at the origin at a later time, under suitable scaling, we prove that the logarithm of the minimizer converges to an explicit function as we tune the value of the conditioning to…
In the present paper we study the minimization of energy integrals on the sphere with a focus on an interesting clustering phenomenon: for certain types of potentials, optimal measures are discrete or are supported on small sets. In particular, we prove that the support of any minimizer of the $p$-frame energy has empty interior whenever $p$ is not an even integer. A similar effect is also demonstrated for energies with analytic potentials which are not positive definite. In addition, we establish the existence of discrete minimizers for a large class of energies, which includes energies with …
In this paper, we focus on finite Bravais lattice energies per point in two dimensions. We compute the first and second derivatives of these energies. We prove that the Hessian at the square and the triangular lattice are diagonal and we give simple sufficient conditions for the local minimality of these lattices. Furthermore, we apply our result to Lennard--Jones type interacting potentials that appear to be accurate in many physical and biological models. The goal of this investigation is to understand how the minimum of the Lennard--Jones lattice energy varies with respect to the density of…
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