William P. Ziemer

Fine Regularity of Solutions of Elliptic Partial Differential Equations

Preliminaries Potential theory Quasilinear equations Fine regularity theory Variational inequalities--Regularity Existence theory References Index Notation index.

Local behavior of solutions of quasilinear elliptic equations with general structure

This paper is motivated by the observation that solutions to certain variational inequalities involving partial differential operators of the form divA(x, u, Vu) + B(x, u, Vu), where A and B are Borel measurable, are solutions to the equation divA(x, u, Vu) + B(x, u, Vu) = ,u for some nonnegative Radon measure ,i. Among other things, it is shown that if u is a Holder continuous solution to this equation, then the measure ,u satisfies the growth property j[B(x, r)] 1 is given by the structure of the differential operator. Conversely, if u is assumed to satisfy this growth condition, then it is shown that u satisfies a Harnack-type inequality, thus proving that u is locally bounded. Under -the additional assumption that A is strongly monotonic, it is shown that u is Holder continuous.

Area minimizing sets subject to a volume constraint in a convex set

For a given convex subset Ω of Euclidean n-space, we consider the problem of minimizing the perimeter of subsets of Ω subject to a volume constraint. The problem is to determine whether in general a minimizer is also convex. Although this problem is unresolved, we show that if Ω satisfies a “great circle” condition, then any minimizer is convex. We say that Ω satisfies a great circle condition if the largest closed ball B contained in Ω has a great circle that is contained in the boundary of Ω. A great circle of B is defined as the intersection of the boundary of B with a hyperplane passing through the center of B.

Pointwise differentiability and absolute continuity

This paper is concerned with the relationships between Lp differentiability and Sobolev functions. It is shown that if f is a Sobolev function with weak derivatives up to order k in Lp, and 0 s I < k, then f has an Lp derivative of order 1 everywhere except for a set which is small in the sense of an appropriate capacity. It is also shown that if a function has an Lp derivative everywhere except for a set small in capacity and if these derivatives are in Lp, then the function is a Sobolev function. A similar analysis is applied to determine general conditions under which the Gauss-Green theorem is valid.

Cauchy flux and sets of finite perimeter

Abstract : A Cauchy flux Q is a real-valued, additive, area-bounded function whose domain is the class of all Borel subsets of the reduced boundary of sets of finite perimeter. If the flux Q is also volume bounded, it is shown that Q can be represented as the integral of the normal component of some vector field. (Author)

The Dirichlet Problem for Functions of Least Gradient

For a given domain Ω⊂ R n, we consider the variational problem of minimizing the L1-norm of the gradient on Ωof a function with prescribed continuous boundary values. Under certain weak conditions on the boundary of the domain Ω, it is shown that the BV solution is continuous and unique.

Functions of Bounded Variation

A function of bounded variation of one variable can be characterized as an integrable function whose derivative in the sense of distributions is a signed measure with finite total variation. This chapter is directed to the multivariate analog of these functions, namely the class of L1functions whose partial derivatives are measures in the sense of distributions. Just as absolutely continuous functions form a subclass of BV functions, so it is that Sobolev functions are contained within the class of BV functions of several variables. While functions of bounded variation of one variable have a relatively simple structure that is easy to expose, the multivariate theory produces a rich and beautiful structure that draws heavily from geometric measure theory. An interesting and important aspect of the theory is the analysis of sets whose characteristic functions are BV (called sets of finite perimeter). These sets have applications in a variety of settings because of their generality and utility. For example, they include the class of Lipschitz domains and the fact that the Gauss-Green theorem is valid for them underscores their usefulness. One of our main objectives is to establish Poincare-type inequalities for functions of bounded variation in a context similar to that developed in Chapter 4 for Sobolev functions. This will require an analysis of the structure of BV functions including the notion of trace on the boundary of an open set.

Sobolev functions whose inner trace at the boundary is zero

Let Ω⊂Rn be an arbitrary open set. In this paper it is shown that if a Sobolev functionf∈W1,p(Ω) possesses a zero trace (in the sense of Lebesgue points) on ϖΩ, thenf is weakly zero on ϖΩ in the sense thatf∈W01,p(Ω).

Mean values of subsolutions of elliptic and parabolic equations

Abstract : Integral averages of weak subsolutions (and supersolutions) in Rn of quasilinear elliptic and parabolic equations are investigated. The important feature is that these integral averages are defined in terms of measures that reflect interesting geometric phenomena. Harnack type inequalities are established in terms of these integral averages. (Author)

The image of a weakly differentiable mapping

Let