A detailed analysis of the class of absolutely minimizing functions in Euclidean spaces and the relationship to the infinity Laplace equation

/pdf/a-tour-of-the-theory-of-absolutely-minimizing-functions-58zy62bl8g.pdf

A tour of the theory of absolutely minimizing functions

We discuss the ill-posed Cauchy problem for elliptic equations, which is pervasive in inverse boundary value problems modeled by elliptic equations. We provide essentially optimal stability results, in wide generality and under substantially minimal assumptions. As a general scheme in our arguments, we show that all such stability results can be derived by the use of a single building brick, the three-spheres inequality.

The stability for the Cauchy problem for elliptic equations

This book provides a quick, but very readable introduction to stochastic differential equations-that is, to differential equations subject to additive "white noise" and related random disturbances. The exposition is strongly focused upon the interplay between probabilistic intuition and mathematical rigour. Topics include a quick survey of measure theoretic probability theory, followed by an introduction to Brownian motion and the Ito stochastic calculus, and finally the theory of stochastic differential equations. The text also includes applications to partial differential equations, optimal stopping problems and options pricing. <br This book can be used as a text for senior undergraduates or beginning graduate students in mathematics, applied mathematics, physics, financial mathematics, etc., who want to learn the basics of stochastic differential equations. The reader is assumed to be fairly familiar with measure theoretic mathematical analysis, but is not assumed to have any particular knowledge of probability theory (which is rapidly developed in Chapter 2 of the book).

/pdf/an-introduction-to-stochastic-differential-equations-4nnzblh52n.pdf

An Introduction to Stochastic Differential Equations

Fractional elliptic equations, Caccioppoli estimates and regularity

http://mate.dm.uba.ar/~smartin/papers/MW%5b8%5d.pdf

A minimum problem with free boundary in orlicz spaces

The regularity theory of free boundaries flourished during the late 1970s and early 1980s and had a major impact in several areas of mathematics, mathematical physics, and industrial mathematics, as well as in applications. Since then the theory continued to evolve. Numerous new ideas, techniques, and methods have been developed, and challenging new problems in applications have arisen. The main intention of the authors of this book is to give a coherent introduction to the study of the regularity properties of free boundaries for a particular type of problems, known as obstacle-type problems. The emphasis is on the methods developed in the past two decades. The topics include optimal regularity, nondegeneracy, rescalings and blowups, classification of global solutions, several types of monotonicity formulas, Lipschitz, $C^1$, as well as higher regularity of the free boundary, structure of the singular set, touch of the free and fixed boundaries, and more. The book is based on lecture notes for the courses and mini-courses given by the authors at various locations and should be accessible to advanced graduate students and researchers in analysis and partial differential equations.

Regularity of Free Boundaries in Obstacle-type Problems

We construct two new one-parameter families of monotonicity formulas to study the free boundary points in the lower dimensional obstacle problem. The first one is a family of Weiss type formulas geared for points of any given homogeneity and the second one is a family of Monneau type formulas suited for the study of singular points. We show the uniqueness and continuous dependence of the blowups at singular points of given homogeneity. This allows to prove a structural theorem for the singular set. Our approach works both for zero and smooth non-zero lower dimensional obstacles. The study in the latter case is based on a generalization of Almgren’s frequency formula, first established by Caffarelli, Salsa, and Silvestre.

https://www.math.purdue.edu/~arshak/pdf/frac-sing-set-rev.pdf

Some new monotonicity formulas and the singular set in the lower dimensional obstacle problem

We study the regularity of the free boundary in a Stefan-type problem 1u − ∂t u = χ in D ⊂ Rn × R, u = |∇u| = 0 on D \ with no sign assumptions on u and the time derivative ∂t u.

https://www.math.purdue.edu/~arshak/pdf/global11.pdf

Regularity of a free boundary in parabolic potential theory

In this paper we prove $C^{1,\alpha}$
 regularity (near flat points) of the free boundary $\partial\{u > 0\}\cap\Omega$
 in the Alt-Caffarelli type minimum problem for the p-Laplace operator: $$J(u)=\int_\Omega\left( |\nabla u|^p + \lambda^p\chi_{\{u>0\}}\right)dx\rightarrow \min\qquad (1<p<\infty).$$

/pdf/a-minimum-problem-with-free-boundary-for-a-degenerate-2cn3w93r02.pdf

A minimum problem with free boundary for a degenerate quasilinear operator

https://www.math.purdue.edu/~arshak/pdf/asplp04v.pdf

Arshak Petrosyan

Papers

Regularity of Free Boundaries in Obstacle-type Problems

Some new monotonicity formulas and the singular set in the lower dimensional obstacle problem

Regularity of a free boundary in parabolic potential theory

A minimum problem with free boundary for a degenerate quasilinear operator

Large-time geometric properties of solutions of the evolution p-Laplacian equation