Nonlinear Partial Differential Equations

我将继续我在非线性偏微分方程方面的研究，重点放在以下几个方面：（i） Hamilton动力学的PDE方法：这里的目标是使用适当单元PDE的（非光滑）解来研究Aubry-Mather集上的Hamilton动力学，用于具有许多自由度的问题。 (ii)最优Lipschitz扩张的正则性："无穷Laplacian“的边值问题是一个高度退化的非线性偏微分方程，这是非常有趣的，因为它是变分问题”超范数“的一种Euler-Lagrange方程。我打算仔细研究弱解的可能正则性。(iii)哈密顿-雅可比方程的松弛近似：形式渐近性强烈地表明，用一个波算子扰动一个哈密顿-雅可比方程，具有足够大的波速，应该在极限下产生一个粘性解。这一点的证明将极大地扩展粘性求解方法的适用性，以涵盖没有最大值原理的近似。 (iv)质量传递问题：我继续对Monge-Kantorovich质量传递问题的PDE方法感兴趣，特别是那些发生在快速时间尺度上的问题，再加上较慢的动力学。偏微分方程（PDE）作为各种各样的物理和其他现象的近似描述而出现，非线性PDE特别困难，因为它们不允许我们将复杂的解分解为简单解的叠加。然而，许多重要的非线性偏微分方程确实具有变分结构，这意味着它们对应于一种优化原理。变分法的总体目标是找到利用这些变分原理的方法，以帮助我们理解相应的非线性偏微分方程解的性质。在这个项目中，我将（大部分）继续研究几个重要的变分问题，以了解更多关于（i）某些非耗散动力系统，（ii）最佳扩展问题，（iii）耗散现象的波动近似，和（iv）最佳质量重新分配问题。

AbstractEvansI will continue my ongoing research in nonlinear PDE, with emphasis on these topics: (i) PDE methods for Hamiltonian dynamics:The goal here is to use the (nonsmooth) solution of the appropriate cellPDE to study Hamiltonian dynamics on the Aubry-Mather set, for problemswith many degrees of freedom. (ii) Regularity for optimal Lipschitz extensions:The boundary value problem for the ``infinity-Laplacian'' is a highlydegenerate nonlinear PDE, which is extremely interesting sinceit is a sort of Euler-Lagrange equation for a calculus of variationsproblem ``in the sup-norm''. I intend to study carefully the possibleregularity of weak solutions. (iii) Relaxation approximations for Hamilton-Jacobi equations: Formal asymptotics strongly suggest that perturbing a Hamilton-JacobiPDE by a wave operator, with large enough wave speed, should in the limitproduce a viscosity solution. A proof of this would dramatically extendthe applicability of viscosity solution methods, to cover approximationswith no maximum principle. (iv) Mass transfer problems:I continue to be interested in PDE methods for Monge-Kantorovich masstransfer problems, especially those occurring on fast time scales, coupled with slower dynamics.Partial differential equations (PDE) occur as mathematicaldescriptions of an extremely wide variety of physical and other phenomena,and nonlinear PDE are especially difficult, since they do not permit us to decomposecomplicated solutions into a superposition of simpler solutions. Many importantnonlinear PDE do however have a variational structure, meaning that they correspond to a sort of optimization principle. The overallgoal of the calculus of varitations is finding ways to exploit these variational principles,to help us understand the nature of solutions to the corresponding nonlinear PDE. In this project I will (mostly) continue to study several important classes of variationalproblems, to understand more about (i) certain nondissipative dynamical systems, (ii) optimal extension problems, (iii) wave-like approximations to dissipativephenomena, and (iv) optimal mass reallocation problems.