Andrew Vogel

Uniqueness in a Free Boundary Problem

We study overdetermined boundary conditions for positive solutions to the p Laplacian in a bounded domain D. We show these conditions imply uniqueness in certain free boundary problems.

A multidirectional Dirichlet problem

B.E.J. Dahlberg’s theorems on the mutual absolute continuity of harmonic and surface measures, and on the unique solvability of the Dirichlet problem for Laplace’s equation with data taken in Lp spaces p > 2 − δ are extended to compact polyhedral domains of ℝn. Consequently, for q < 2 + δ, Dahlberg’s reverse Hölder inequality for the density of harmonic measure is established for compact polyhedra that additionally satisfy the Harnack chain condition. It is proved that a compact polyhedral domain satisfies the Harnack chain condition if its boundary is a topological manifold. The double suspension of the Mazur manifold is invoked to indicate that perhaps such a polyhedron need not itself be a manifold with boundary; see the footnote in Section 9. A theorem on approximating compact polyhedra by Lipschitz domains in a certain weak sense is proved, along with other geometric lemmas.

On Some Almost Everywhere Symmetry Theorems

In this note we consider problems of the following type: Let D be a bounded domain in Euclidean n space (R n ) and suppose u is a positive solution to Lu = 0 in D with u(x) → 0, |Δu(x)| → constant, as x → ∂D, in an appropriate sense. Show that D is a ball and u is radially symmetric about the center of D. This problem was solved very elegantly by Serrin [25] under the assumption that ∂D is of class C 2, and where a, b, c, are continuously differentiable in each variable. Also L is elliptic and repeated indices denote summation from 1 to n. Immediately following Serrin’s article, Weinberger [30] gave another proof of Serrin’s theorem when Lu = Δu +1 = 0. From Weinberger’s proof it is clear that the assumption, ∂D ∈ C 2 is unnecessary in this special case, provided the boundary conditions are interpreted as, (A) Given e > 0 there exists a neighborhood N of ∂D such that u(x) < e, |∇u(x)|—a| < e, when x ∈ N. In (A), a denotes a positive constant.

On pseudospheres that are quasispheres

We construct bounded domains D not equal to a ball in n = 3 dimensional Euclidean space, Rn, for which ?D is homeomorphic to a sphere under a quasiconformal mapping of Rn and such that n - 1 dimensional Hausdorff measure equals harmonic measure on ?D.

On the Dimension of p-Harmonic Measure in Space

Let R n , n 3, and let p, 1 0 small such that if is a -Reifenberg flat domain with < ˜ , then p-harmonic measure is concentrated on a set of -finite H n 1 measure. The situation is more interesting when 1 < p < n as we show by examples involving certain Wol snowflakes. Our results complement work of the first three authors in R 2 (along with Pietro Poggi-Corradini), see [BL], [L] and [LNP], where similar results for the dimension p-harmonic measure were obtained in a Jordan domain bounded by a quasicircle and in simply connected domains.

Symmetry Theorems and Uniform Rectifiability

We study overdetermined boundary conditions for positive solutions to some elliptic partial differential equations of-Laplacian type in a bounded domain. We show that these conditions imply uniform rectifiability of and also that they yield the solution to certain symmetry problems.

A symmetry theorem revisited

We show that if harmonic measure and Hausdorff measure are equal on the boundary of certain domains in Euclidean n-space, then these domains are necessarily balls.