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Let Ω be a bounded, convex, open subset of R N . This paper concerns the behavior of positive harmonic functions that vanish on ∂Ω. We will consider both the case in which the function is defined in Ω and the case in which the function is defined in the complement of Ω. We will also discuss eigenfunctions defined in Ω. The theme is to study how the shape of Ω influences the size of solutions to these basic elliptic equations.

Consumer expenditure data often violate the revealed preference axioms. The observed revealed preference inconsistencies must be measured in order to judge the extent to which optimizing models misrepresent demand behavior. This paper considers the “sizes” of violations of the Ville axiom, a differential revealed preference axiom that is equivalent to Slutsky symmetry. We study the effect of commodity aggregation on Ville axiom violations when the commodity aggregation is justified in the sense that it generates a well-defined aggregated demand function. Using a natural measure of Slutsky asymmetry and Ville axiom violation, we show that commodity aggregation can either expand or eliminate the revealed preference inconsistency. We obtain an upper bound on the increase in the maximal sizes of Ville axiom violations due to commodity aggregation. The Slutsky asymmetry norm of the aggregated demand can be larger than that of the original demand function by a factor that is no greater than \( \sqrt {nm/N,} \), where N is the number of goods in the original demand system, m is the number of goods that are aggregated, and n = N − m + 1 is the number of goods in the aggregated system.

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.

In this note we will prove estimates for harmonic measure in convex and convex C 1 domains. It is not hard to show that in a convex domain, surface measure belongs to the Muckenhoupt class A 1 with respect to harmonic measure (Lemma 3). If the boundary of the domain is also of class C 1, then it follows from [JK1] that the constant in the A 1 condition tends to 1 as the radius of the ball tends to 0 (Lemma 7′). Our main estimates (Theorems A and B) are of the same type. The novelty is that they are not calculated with respect to balls, but rather with respect to “slices” formed by the intersection of the boundary with an arbitrary half-space. In addition to proving Theorems A and B we will also explain how these estimates are related to an approach to regularity for the Monge-Ampere equation due to L. Caffarelli and to a problem of prescribing harmonic measure as a function of the unit normal.

We extend the two-dimensional results of Jerison (2000) on the location of the nodal set of the first Neumann eigenfunction of a convex domain to higher dimensions. If a convex domain Ω in ℝ n is contained in a long and thin cylinder [0,N] x B ∈ (0) with nonempty intersections with {x 1 = 0} and {x 1 = N}, then the first nonzero eigenvalue is well approximated by the eigenvalue of an ordinary differential equation, by a bound proportional to ∈, whose coefficients are expressed in terms of the volume of the cross sections of the domain. Also, the first nodal set is located within a distance comparable to e near the zero of the corresponding ordinary differential equation.

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

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

On demontre les meilleures inegalites de type Carleman possibles dans le cas de l'operateur de Dirac. On obtient la propriete de prolongement unique pour D+V, ou V∈L loc γ(R n ), γ=(3n-2)/2, (n≥3)

We find indirect utility and expenditure functions and the implications of Slutsky symmetry for the HOGLEX demand function X ( p , y ) = y α ( p ) B ( p )+ yC ( p ). These implications restrict the usefulness of the HOGLEX function in the theory of preference aggregation unless α is constant.

We obtain a collection of multilinear Littlewood-Paley estimates, which we then apply to two problems in partial differential equations. The first problem is the estimation of the square root of an elliptic operator in divergence form, and the second is the estimation of solutions to the Cauchy problem for nondivergence-form parabolic equations.

A CR structure on a real manifold M is a distinguished complex subbundle Z of the complex tangent bundle C TM, satisfying X n Z = 0 and [k, X] c X. For example, the complex structure of Cn+I induces a natural biholomorphically invariant CR structure on any real hypersurface: *' is the space of vectors in the span of a/lz1, . /. ., aaZn+l which are tangent to the hypersurface. An abstract CR manifold M is said to be of hypersurface type if dimR M = 2n + I and dim, X = n; all our CR manifolds will be of this type. If M is oriented, then there is a globally defined real one-form 0 that annihilates *' and . The Levi form, given by Lo (V, W) = -2idO (V A W), is a hermitian form on X. We will assume that the CR structure is strictly pseudoconvex: for some choice of 0, the Levi form Lo is positive definite on Z. In this case 0 defines a contact structure on M and we call 0 a contact form associated with the CR structure. The Levi form plays a role similar to that of the metric in Riemannian geometry. However, the CR structure only determines the Levi form up to a conformal multiple; this multiple is fixed by the choice of a contact form. A CR structure with a given choice of contact form is called a pseudohermitian structure. Thus there is an analogy between pseudohermitian and CR manifolds on the one hand and Riemannian and conformal manifolds on the other. In particular, Webster [W1, W2] and Tanaka [T] have defined a pseudohermitian scalar curvature associated to Lo. The CR Yamabe problem is: given a compact, strictly pseudoconvex CR manifold, find a choice of contact form for which the pseudohermitian scalar curvature is constant. Suppose M is a strictly pseudoconvex CR manifold of dimension 2n + 1. Solutions to the CR Yamabe problem are precisely the critical points of the CR

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

Two theorems are given regarding the means of quasiconformal and quasiregular mappings. Together they show that the principle of subordination for means of analytic functions has no analog, at least in the case of plane quasiregular mappings.