Lavi Karp

Regularity of a free boundary with application to the Pompeiu problem

In the unit ball B(0; 1), let u and › (a domain in N ) solve the following overdetermined problem: ¢u = ´› in B(0; 1); 02 @› ;u =jruj =0 inB(0; 1)n ›; where ´› denotes the characteristic function, and the equation is satisfled in the sense of distributions. If the complement of › does not develop cusp singularities at the origin then we prove @› is analytic in some small neighborhood of the origin. The result can be modifled to yield for more general divergence form operators. As an application of this, then, we obtain the regularity of the boundary of a domain without the Pompeiu property, provided its complement has no cusp singularities.

Regularity of a free boundary problem

Let u be the Newtonian potential of a real analytic distribution in an open set Ω. In this paper we assume u is analytically continuable from the complement of Ω into some neighborhood of a point x0 ∈ ∂Ω, and we study conditions under which the analytic continuation implies that ∂Ω is a real analytic hypersurface in some neighborhood of x0.

On the newtonian potential of ellipsoids

It is a well known fact that the Newtonian potential of a uniform mass distribution in an ellipsoid is equal to a quadratic polynomial inside the ellipsoid. Conversely, if K is a bounded solid in R m and its Newtonian potential is equal to a quadratic polynomial inside it, then K is an ellipsoid. P. Dive proved this result for m=3 in 1931, and in 1986 E. DiBenedetto and A. Friedman showed it to be true for all m ≥ 2. We use certain topological methods to obtain a simpler proof of this result.

Construction of quadrature domains in R nfrom quadrature domains in R 2

We construct some quadrature domains (q.d.) in R 4, and R 3 from q.d. in R 2. We obtain a q.d. in R 4 by taking a certain symmetric q.d. D⊂R 2, and rotating it in R 4. For a q.d. D ⊂ R 2 whose corresponding distribution T has an order ≥ 1, we show that Ω = D×R 1 is a q.d. in R 3, with distribution equal to T&dx⊗3.

On Null Quadrature Domains

The characterization of null quadrature domains in Rn, n ≥ 3, has been an open problem throughout the past two and a half decades. A substantial contribution was made by Friedman and Sakai [10]; they showed that if the complement is bounded, then null quadrature domains are exactly the complement of ellipsoids. The first result with unbounded complements appeared in [16], there it is assumed the complement is contained in an infinitely cylinder.The aim of this paper is to show the relation between null quadrature domains and Newton’s theorem on the gravitational force induced by homogeneous homoeoidal ellipsoids. We also make progress in the classification problem and we show that if the boundary of null quadrature domain is contained in a slab and the complement satisfies a certain capacity condition at infinity, then it must be a half-space or a complement of a slab. In addition, we present a Phragmén-Lindelöf type theorem which we could not find in the literature.

Regularity of a free boundary at the infinity point

Suppose there is a nonnegative function u and an open set , satisfying where . Under a certain thickness condition on Rn/Ω, we prove that the boundary of is a graph of a C1function in a neighborhood of the origin. As a by-product of the method of the proof, we also obtain the following result: Replace xωby fxωwith a certain assumptions on f. Then for any solution u which is asymptotically nonnegative at infinity, there holds

Isolated Singularities of Harmonic Functions

The main result of this paper gives a sufficient condition for removability of an isolated singularity of a harmonic function. The condition is given in terms of Newtonian capacity. In addition, an application to an approximation problem is presented. Introduction This note deals with a problem of removable isolated singularities of harmonic functions. Suppose u is harmonic and of at most polynomial growth on a punctured neighborhood of a point x0 and its gradient ∇u satisfies |∇u(x)| ≤ C|x − x0|−1 on a set K, where x0 is an accumulation point of K. Under these hypotheses, what size restrictions on K ensure that x0 is a removable singularity of u? This type of growth constraint is rather natural in the case of harmonic functions, while for analytic functions, a bound on the function is often used, and the answer in this case is very well known (see e.g. [1], Proposition 2.4.4): Theorem A. Assume f is holomorphic and single valued on the punctured disk {z : 0 < |z − z0| < r} and bounded on a sequence of points clustering at z0. Then f cannot have a polar singularity at z0. The two dimensional case of the removable singularities problem for harmonic functions mentioned above, and with a bounded gradient on the set K, is settled by Theorem A. However, in higher dimensions the situation is more complicated. We present here a generalization of the above for harmonic functions in IR , n ≥ 3. The Newtonian capacity will measure the ”size” of K. In order to state our result, we introduce some notations. We let Br(x0) = {x : |x − x0| < r} be the ball of radius r centered at x0, Br = Br(0), S(a, b, x0) = {x : a ≤ |x − x0| ≤ b} and S(a, b) shall denote S(a, b, 0). The 2 L. Karp, H.S. Shapiro gradient of a function is denoted by ∇, ∆ is the Laplace operator and cap (K) denotes the Newtonian capacity of the set K (see [4]). Theorem 0.1. Let m be a positive integer and assume that a function u satisfies (a) ∆u = 0 in B1(x0) \ {x0}, (b) u(x) = O(|x− x0|−m) as x → x0, and (c) |∇u(x)| ≤ C |x−x0| on K ∩ (B1(x0) \ {x0}). If lim sup r→0 cap (