Eugene B. Fabes

On a regularity theorem for weak solutions to transmission problems with internal Lipschitz boundaries

We show that if u is a weak solution to div(A⊇u)=0 on an open set Ω containing a Lipschitz domain D, where A=kI χD+ I χΩ/D (k>0, k¬=1), then the nontangential maximal function of the gradient of u lies in L 2 (∂D)

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

Construction of inertial manifolds by elliptic regularization

Abstract In many cases an inertial manifold 2 M for an infinite dimensional dissipative dynamical system can be represented as the graph of a smooth function Φ from a finite dimensional Hilbert space Hp to another Hilbert space Hq. The invariance property of M means that Φ can be written as the solution of a first order partial differential equation DΦ(p)G1(p, Φ(p)) + AΦ(p) = G2(p, Φ(p)) (0) over Hp, where G1 and G2 are nonlinear functions which depend on the original dynamical system and A is a suitably “stable” linear operator. In this paper we use a method introduced by Sacker (R. J. Sacker, A new approach to the perturbation theory of invariant surface, Comm. Pure Appl. Math.18 (1965), 717–732), for the study of finite dimensional dynamical systems, to find inertial manifolds in the infinite dimensional setting. This method involves replacing the first order equation for Φ by the regularized elliptic equation −eΔΦ + DΦ(p) G1,(p, Φ(P)) + AΦ(p) = G2(p, Φ(p)), with suitable boundary conditions. It is shown that if A satisfies a spectral gap condition, then the solutions Φe of the elliptic equation converge to a weak solution Φ of (0), as e → 0+. Furthermore, M = Graph Φ is an invariant manifold for the given dynamical system.

The inverse conductivity problem with one measurement: uniqueness for convex polyhedra

Let L denote a smooth domain in Rn containing the closure of a convex polyhedron D. Set XD equal to the characteristic function of D. We find a flux g so that if u is the nonconstant solution of div ((1 + XD)VU) = 0 in Q with Ou = g on OQ, then D is uniquely determined by the Cauchy On data g and f u/Al. INTRODUCTION Let Q be a bounded domain in Rn, n > 2, with a connected boundary and D a subdomain in Q . Assume both Q and D are conductors of electricity. We consider the following question: Can we set up a magnetic field E surrounding Q with a known flux g across aQ so that calculating the potential of the field on aQ will determine D? Writing E = Vu we have Ly u= div(y(x)Vu) = O in Q, au g onOQ, where y(x) is the conductivity and flu denotes the normal derivative of u on AQ. The question now becomes: Can we choose g so that g and f = ulja uniquely determine D? For the sake of definiteness we take y(x) = 1 + XD(X), where XD denotes the characteristic function of D. In this case, Friedman and Isakov [1] proved that there is a flux g so that g and f uniquely determine D if D is assumed to be a convex polyhedron situated away from the boundary of Q, that is, if diam(D) < dist(D, AQ). In this paper we are able to remove this extra condition. Specifically, we prove the following uniqueness result: There exists a function g defined on Received by the editors August 19, 1992 and, in revised form, December 10, 1992; contents presented at an international meeting titled Partial Differential Equations of Elliptic Type held in Cortona, Italy, October 12-16, 1992, and sponsored by Istituto Nazionale di Alta Matematica and Consiglio Nazionale delle Ricerche. 1991 Mathematics Subject Classification. Primary 35R30, 35R05, 35B30. Q 1994 American Mathematical Society 0002-9939/94

Weak-type estimates for the Riesz transforms associated with the gaussian measure

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Uniqueness in the dirichlet problem for some elliptic operators with discontinuous coefficients

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A partial answer to a conjecture of B. Simon concerning unique continuation

In this paper we will study the behavior of the Riesz transform associated with the Gaussian measure ?(x)dx = e-|x|2dx in the space L?1 (Rn).

The Spectral Radius of the Classical Layer Potentials on Convex Domains

Uniqueness is proved for the Dirichlet problem for second order nondivergence form elliptic operators with coefficients continuous except at a countable set of points having at most one accumulation point. Moreover, gradient estimates are proved.

Inverse conductivity problem with one measurement: error estimates and approximate identification for perturbed disks

In his survey paper on Schrijdinger semigroups [S] B. Simon formulated the following Conjecture (see [S, p. 5193). Let VE KfP”, the local Kato class of potentials. Then the time-independent Schrddinger operator H= -d + V has the unique continuation property (ucp). By this it is meant that given any connected open subset 52 c R”, the only solution u of Hu = 0 in 52 that can vanish in an open subset 52, c Sz is u 0. A measurable function V on R” is said to belong to the Kato class K,, if

On the spectra of a Hardy kernel

Let D denote a bounded Lipschitz domain in R n. For almost every (with respect to surface measure dσ)Q ∈∂D the exterior normal N Q at Q exists. The solution u to the Dirichlet problem,