# Your search: "author:"Evans, Lawrence C.""

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## Scholarly Works (12 results)

Viscosity Solution Methods in Risk Analysis

This dissertation is a study of second order, elliptic partial differential equations (PDE) that subject solutions to pointwise gradient constraints. These equations fall into the broad class of scalar non-linear PDE, and therefore, we interpret solutions in the viscosity sense and use methods from the theory of viscosity solutions. These equations are also naturally associated to free boundary problems as the boundary of the region where the gradient constraint is strictly satisfied cannot, in general, be determined before a solution of the PDE has been obtained. Consequently, we also employ techniques from PDE theory developed for free boundary problems.

In addition, we identify connections with control theory. Each solution of the PDE we consider has a probabilistic interpretation as an optimal value of a stochastic control problem. A distinguishing feature of these optimization problems is that the controlled processes have sample paths of bounded variation and thus may be ``singular" with respect to Lebesgue measure on the real line. The theory of stochastic singular control has been used to model spacecraft control, queueing systems, and financial markets in the presence of transaction costs. Our work makes considerable progress at rigorously interpreting the PDE that arise in these applications.

In this thesis, we explore some mathematical properties of the electro-chemical model for lithium-ion batteries, which is a nonlinear system of algebraically coupled diffusion equations. We prove a local existence and uniqueness theorem. Furthermore, we prove the asymptotic stability of a modified version, the so called Single Particle Model for several materials.

In this thesis we study variational inequalities with gradient constraints. We consider the questions of regularity of their solutions, and regularity of their free boundaries. We also study the relation between variational inequalities with gradient constraint and the obstacle problem. In particular we consider their relation in the vector-valued case.

In Chapter 2 we generalize some known results about the equivalence of variational inequalities with gradient constraint and variational inequalities with constraint on the so- lution, by allowing less regular functionals and constraints. In Chapter 3 we prove that some class of vector-valued variational inequalities with gradient constraint is equivalent to variational inequalities with constraint on the solution.

In Chapter 4 we prove the optimal regularity for a class of variational inequalities with gradient constraints. The regularity and the shape of the free boundary of this problem is the subject of Chapters 5 and 6. We prove that the free boundary is as regular as the part of the boundary that parametrizes it. In order to prove this, we generalize the notion of ridge, by replacing the Euclidean norm by other norms. We also consider the singularities of the distance function (in the new norm) to the boundary of a domain.

We study nonlinear partial differential equations describing Hele-Shaw flows for which the source of the fluid, which classically is at a single point, instead expands as the moving boundary uncovers new sources. This problem is reformulated as a type of quasi- variational inequality, for which we show there exists a unique weak solution. Our procedure utilizes a Baiocchi integral transform. However, unlike in traditional applications, the inequality solved by the transformed variable continues to depend on the free boundary, and thus standard theory of variational inequalities cannot be immediately applied. Instead, we develop convergent time-stepping scheme. As a further application, we generalize and study a related problem with a growing density function obeying a hard constraint on the maximum density.

In this dissertation, I provide a fairly simple and direct proof of the asymptotics of the scaled Kramers-Smoluchowski equation in one dimension. I further generalize this result to the cases where the potential function H (1) has three or more wells, (2) is periodic, and (3) has infinitely many wells.

This thesis presents novel methods for computing optimal pre-commitment strategies in time-inconsistent optimal stochastic control and optimal stopping problems. We demonstrate how a time-inconsistent problem can often be re-written in terms of a sequential optimization problem involving the value function of a time-consistent optimal control problem in a higher-dimensional state-space. In particular, we obtain optimal pre-commitment strategies in a non-linear optimal stopping problem, in an optimal stochastic control problem involving conditional value-at-risk, and in an optimal stopping problem with a distribution constraint on the admissible stopping times. In each case, we relate the original problem to auxiliary time-consistent problems, the value functions of which may be characterized in terms of viscosity solutions of a Hamilton-Jacobi-Bellman equation.

I present two recent research directions in this dissertation. The first direction is on the study of the Nonlinear Adjoint Method for Hamilton-Jacobi equations, which was introduced recently by Evans [27]. The main feature of this new method consists of the introduction of an additional equation to derive new information about the solutions of the regularized Hamilton-Jacobi equations. More specifically, we linearize the regularized

Hamilton-Jacobi equations first and then introduce the corresponding adjoint equations. Looking at the behavior of the solutions of the adjoint equations and using integration by

parts techniques, we can prove new estimates, which could not be obtained by previous techniques.

We use the Nonlinear Adjoint Method to study the eikonal-like Hamilton-Jacobi equations [79], and Aubry-Mather theory in the non convex settings [10]. The latter one is based on joint work with Filippo Cagnetti and Diogo Gomes. We are able to relax the

convexity conditions of the Hamiltonians in both situations and achieve some new results.

The second direction, based on joint work with Hiroyoshi Mitake [69, 68], concerns the study of the properties of viscosity solutions of weakly coupled systems of Hamilton-Jacobi equations. In particular, we are interested in cell problems, large time behavior, and homogenization results of the solutions, which have not been studied much in the literature.

We obtain homogenization results for weakly coupled systems of Hamilton-Jacobi equations with fast switching rates and analyze rigorously the initial layers appearing naturally in the systems. Moreover, some properties of the effective Hamiltonian are also derived.

Finally, we study the large time behavior of the solutions of weakly coupled systems of Hamilton-Jacobi equations and prove the results under some specific conditions. The general case is still open up to now.

We study a single risky financial asset model subject to price impact and transaction cost over infinite and finite horizon. An investor needs to execute a long position in the asset affecting the price of the asset and possibly incurring in fixed transaction cost. The objective is to maximize the discounted revenue obtained by this transaction. This problem is formulated first as an impulse control problem and we characterize the value function using the viscosity solutions framework. We establish an associated optimal stopping problem that provides bounds and in some cases the solution of the value function. We also analyze the case where there is no transaction cost and how this formulation relates with a singular control problem. A viscosity solution characterization is provided in this case as well. An explicit solution of the value function is calculated in a particular case. Numerical examples with different types of price impact conclude the discussion.