Henrik Shahgholian

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 properties of a free boundary near contact points with the fixed boundary

In the upper half of the unit ball B+ = {\x\ 0}, let u and Omega (a domain in R-+(n) = {X is an element of R-n : x(1) > 0}) solve the following overdetermined problem: Deltau = chi(Omega) in B+, u ...

The one phase free boundary problem for the p-Laplacian with non-constant Bernoulli boundary condition

Our objective, here, is to generalize our earlier results on the existence of classical convex solution to a free boundary problem with a Bernoulli-type boundary gradient condition and with the p-Laplacian as the governing operator. The main theorems of this paper assert that the exterior and the interior free boundary problem with a Bernoulli law, i.e. with a prescribed pressure a(x) on the free streamline of the flow, have convex solutions provided the initial domains are convex. The continuous function a(x) is subject to certain convexity properties. In our earlier results we have considered the case of constant a(x). In the lines of the proof of the main results we also prove the semi-continuity (up to the boundary) of the gradient of the p-capacitary potentials in convex rings, with C-1 boundaries.

On quadrature domains and the Schwarz potential

Abstract Let T be a distribution in R n whose support is compact. A domain Ω in R n is said to be a quadrature domain with respect to T if supp T ⊂ Ω and ∫ Ω h = 〈h, T〉 , for all functions h which are harmonic and integrable over Ω. We denote this by Ω ϵ Q(T, HL 1 ) . This is equivalent to saying that there is a solution u (called the modified Schwarz potential) to the following overdetermined Cauchy problem: Δu = 1 − T in Ω, and u = ¦▽u¦ = 0 on ∂Ω . In this paper we investigate different properties of quadrature domains. We show that if T is a fixed distribution, then bounded domains in Q(T, HL1) are uniformly bounded. Later we prove that if the modified Schwarz potential of a domain Ω is a polynomial, then Ω is a half-space. We also study the uniqueness problem. Namely, we show that if D and Ω are in Q(T, HL1), v and u are the MSPs of D and Ω, respectively, and sup∂D u ⩾ 0, then ∂Ω meets {x ϵ Ω: v(x) . Consequently if v ⩾ 0, then ¦Ω Δ D¦ = 0 .

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.

A characterization of the sphere in terms of single-layer potentials

Let Q be a bounded smooth domain in Rn, and suppose the singlelayer potential of aQ coincides for y 0 Q with the function clyI-I (c > 0) . Then aQ is a sphere centered at the origin. Throughout this paper we assume Q c 23 (the case Rn is similar) is a bounded domain, and the boundary aQ is smooth enough that for (1) JAQ~~u(y IX Y VY E R one has a u au (2) 4-r Ot onaQ, where dau denotes the surface measure, + indicates the limit from the exterior and the limit from the interior (see [K, p. 164]), and n is the outward normal vector to a Q, which we assume to exist. If Q has this property, then we say Q is smooth. Theorem. Let Q be a bounded smooth domain in R3, and suppose for some c > O (3) jda -,, Vy

Quadrature surfaces as free boundaries

4Q. (3) /9~~~~~Q 1X Y IYI Then, aQ is a sphere centered at the origin. Proof. Since the single-layer potential is continuous in 23 (see [K, p. 160]) (3) implies 0 E Q. Therefore we can take two balls B1, B2 both centered at the origin, such that B1 is the largest ball in Q and B2 is the smallest ball containing Q. Now the idea is to show that aBI = aB2 = aQ. Now let y E aB1 n Q and y2 E aB2 n OQ, then 1y21 > ylvI. Thus it suffices to prove Iy221 < Iyl v . Define u to be the function represented by ( 1). Then by the maximum principle (since u is harmonic in Q) max u in Q is attained on Received by the editors January 7, 1991. 1980 Mathematics Subject Classification (1985 Revision). Primary 3 1B20.

Free boundary regularity close to initial state for parabolic obstacle problem

AbstractThis paper deals with a free boundary porblem connected with the concept “quadrature surface”. Let Ω⊂Rn be a bounded domain with aC2 boundary and μ a measure compactly supported in Ω. Then we say ∂Ω is a quadrature surface with respect to μ if the following overdetermined Cauchy problem has a solution.

Applications of Fourier Analysis in Homogenization of the Dirichlet Problem: L p Estimates

\Delta u = - \mu in \Omega ,u = 0 and \frac{{\partial u}}{{\partial v}} = - 1 on \partial \Omega .