Forced Oscillations for Lagrangian and Hamiltonian Systems and the Nonhomogeneous Dirichlet Problem for Semilinear Elliptic Equations Involving Critical Exponents

在这个项目期间，将对微分方程组领域的几个问题进行研究。它们涉及物理科学中的微分方程组，即表示机械系统强迫运动的拉格朗日方程和更一般的哈密顿系统。强迫摆系统将产生第一类方程，而物体在重力作用下的运动通常使用哈密顿结构。本文所涉及的拉格朗日方程具有周期系数和强迫函数。在这样的假设下，人们很自然地会问是否存在周期解。与许多模拟物理世界现象的方程一样，拉格朗日是由能量或作用量最小化原则产生的。求取自Hilbert空间的允许函数的某一积分的临界点。根据势的增长，可能会有许多不同的周期解，而有证据表明，对于周期势，可能只存在有限多个周期解。将寻求从潜力结构的角度衡量这类解决办法的数量。对哈密顿系统的研究将遵循类似的思路，集中在周期哈密顿系统的情况下。这些系统的解称为次谐波。众所周知，在非常一般的条件下，存在周期次谐波。最重要的次谐波是那些周期最短的次谐波。这项研究将在对哈密顿量作额外的对称性假设的情况下，寻找具有最小周期的解的描述。哈密顿系统的解必须位于恒定能量的曲面上。这项工作的一个相关目标将是确定哈密顿的哪些表面带有周期轨道。

Several problems in the field of differential equations will be under investigation during the term of this project. They concern systems of differential equations from the physical sciences, namely Lagrangian equations representing forced motion of a mechanical system and the more general Hamiltonian systems. A system of forced pendulums would give rise to the first class of equations, whereas the motion of bodies under gravitational forces often uses the Hamiltonian structure. The Lagrangian equations involved in this study have periodic coefficients and forcing functions. It is natural to inquire whether of not there will exist periodic solutions under such assumptions. As with many equations modeling phenomena from the physical world, the Lagrangians result from a principle of energy or action minimization. One seeks to find critical points of a certain integral with admissible functions taken from a Hilbert space. Depending on the growth of the potential, there may be many distinct periodic solutions, whereas evidence suggests that for periodic potentials, only finitely many periodic solutions may exist. A measure of the number of such solutions in terms the structure of the potential will be sought. Work on Hamiltonian systems will follow a similar vein concentrating on the case of periodic Hamiltonians. Solutions of these systems are known as subharmonics. There are known to be periodic subharmonics under very general conditions. The most important subharmonics are those with minimal periods. This research will look for a description of solutions with minimal periods in cases where additional symmetry assumptions are made on the Hamiltonian. Solutions of Hamiltonian systems must lie on surfaces of constant energy. A related goal of this work will be to determine which surfaces of the Hamiltonian carry a periodic orbit.