David Jerison

The Neumann problem on Lipschitz domains

Au — 0 in D; u = ƒ on bD9 where ƒ and its gradient on 3D belong to L(do). For C domains, these estimates were obtained by A. P. Calderón et al. [1]. For dimension 2, see (d) below. In [4] and [5] we found an elementary integral formula (7) and used it to prove a theorem of Dahlberg (Theorem 1) on Lipschitz domains. Unknown to us, this formula had already been discovered long ago by Payne and Weinberger and applied to the Dirichlet problem in smooth domains. Moreover, they used a second formula (2), which is a variant of a formula due to F. Rellich [7], to study the Neumann problem in smooth domains. We show here that the same strategy as in [4] applied to the second formula (2) coupled with Dahlbergs theorem yields our main result. Thus integral formulas give appropriate estimates for the solution of not only the Dirichlet problem, but also the Neumann problem on Lipschitz domains. We will present a more general version that applies to variable coefficient operators, systems, and other elliptic problems in a later

Some new monotonicity theorems with applications to free boundary problems

The monotonicity theorem of Alt, Caffarelli, and Friedman [ACF, Lemma 5.1] plays a central role in the existence and regularity theory of two-phase free boundary problems. It has been extended, for example, to variable coefficient operators [C3, Lemma 1] and to eigenvalue problems [FL]. In this paper we reformulate the theorem so that it applies to inhomogeneous equations in which the right-hand side of the equation need not vanish at the free boundary. Of particular interest is the case in which Au takes two different constant values on the set where u > 0 and on the set where u < 0. The Prandtl-Batchelor

Necessary and sufficient conditions for absolute continuity of elliptic-harmonic measure

Our purpose in this paper is to give a necessary and sufficient condition on the modulus of continuity of the coefficients of an elliptic operator in divergence form in order that the corresponding Poisson kernel for the Dirichlet problem exist. This condition on the coefficients is global continuity together with the additional property that the modulus of continuity along some nontangential direction at each boundary point be bounded uniformly in these points and directions by a function 7(t) satisfying the Dini-type condition

A singular energy minimizing free boundary

Abstract We consider the problem of minimizing the energy functional ∫(|∇u|2 + χ {u>0}). We show that the singular axissymmetric critical point of the functional is an energy minimizer in dimension 7. This is the first example of a non-smooth energy minimizer. It is analogous to the Simons cone, a least area hypersurface in dimension 8.

Logarithmic Fluctuations for Internal DLA

Let each of n particles starting at the origin in Z^2 perform simple random walk until reaching a site with no other particles. Lawler, Bramson, and Griffeath proved that the resulting random set A(n) of n occupied sites is (with high probability) close to a disk B_r of radius r=\sqrt{n/\pi}. We show that the discrepancy between A(n) and the disk is at most logarithmic in the radius: i.e., there is an absolute constant C such that the following holds with probability one: B_{r - C \log r} \subset A(\pi r^2) \subset B_{r+ C \log r} for all sufficiently large r.

The size of the first eigenfunction of a convex planar domain

The goal of this paper is to estimate the size of the first eigenfunction u uniformly for all convex domains. In particular, we will locate the place where u achieves its maximum to within a distance comparable to the inradius, uniformly for arbitrarily large diameter. In addition, we will estimate the location of other level sets of u by showing that u is well-approximated by the first eigenfunction of a naturally associated ordinary differential (Schrodinger) operator. We intend to show in a separate paper that the estimates here are best possible in order of magnitude. The present paper depends on the ideas and results of our earlier work [J] and [GJ], where detailed estimates for the zero set of the second eigenfunction (or first nodal line) are obtained. The paper [J] also contains some estimates for the first eigenfunction and lowest eigenvalue, but the techniques of [GJ] and new techniques introduced here are essential to the best possible estimates for the first eigenfunction presented here. The maximum of the first eigenfunction occurs at the point of largest displacement of a vibrating drum with fixed edges when it vibrates at its fundamental or first resonant frequency. The first nodal line is the stationary curve of the drum at the second resonant frequency. The maximum is harder to find experimentally than the nodal line because it is a single point. Its location has less influence on the eigenvalue or Dirichlet integral, so it is also harder to locate mathematically. Another way to describe the difficulty is as follows. To find the maximum of the first eigenfunction we will need to estimate its first directional derivative. Derivatives of the first eigenfunction are, roughly speaking, analogous to the second eigenfunction because they are solutions to an eigenfunction equation. Moreover, convexity properties of the first eigenfunction imply that the zero set of the derivative divides the region into two connected components. But the derivatives are harder to estimate than a second eigenfunction because they do not vanish at the boundary.

The Diameter of the First Nodal Line of a Convex Domain

The main goal of this paper is to prove that the first nodal line for the Dirichlet problem in a convex planar domain has diameter less than an absolute constant times the inradius of the domain. More precisely, we locate the nodal line, to within a distance comparable to the inradius, near the zero of an ordinary differential equation, which is associated to the domain in a natural way. We also derive estimates for the first and second eigenvalues in terms of the corresponding eigenvalues of the ordinary differential equation and construct an approximate first eigenfunction. Two examples, a rectangle and a circular sector, illustrate the two extreme possibilities for the location of the nodal line. For the rectangle R = { (x, y) o 1, we have the second eigenfunction

Locating the first nodal linein the Neumann problem

The location of the nodal line of the first nonconstant Neumann eigenfunction of a convex planar domain is specified to within a distance comparable to the inradius. This is used to prove that the eigenvalue of the partial differential equation is approximated well by the eigenvalue of an ordinary differential equation whose coefficients are expressed solely in terms of the width of the domain. A variant of these estimates is given for domains that are thin strips and satisfy a Lipschitz condition.

Regularity for the one-phase Hele-Shaw problem from a Lipschitz initial surface

In this paper we show that if the Lipschitz constant of the initial free boundary is small, then for small positive time the solution is smooth and satisfies the Hele-Shaw equation in the classical sense. A key ingredient in the proof which is of independent interest is an estimate up to order of magnitude of the speed of the free boundary in terms of initial data.

Partial Differential Equations with Minimal Smoothness and Applications

In recent years there has been a great deal of activity in both the theoretical and applied aspects of partial differential equations, with emphasis on realistic engineering applications, which usually involve lack of smoothness. On March 21-25, 1990, the University of Chicago hosted a workshop that brought together approximately fortyfive experts in theoretical and applied aspects of these subjects. The workshop was a vehicle for summarizing the current status of research in these areas, and for defining new directions for future progress - this volume contains articles from participants of the workshop.