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Ultrasound modulation of electrical or optical properties of materials offers the possibility of devising hybrid imaging techniques that combine the high electrical or optical contrast observed in many settings of interest with the high resolution of ultrasound. Mathematically, these modalities require that we reconstruct a diffusion coefficient for , a bounded domain in , from knowledge of for , where is the solution to the elliptic equation in with on .
This inverse problem may be recast as a nonlinear equation, which formally takes the form of a -Laplacian. Whereas -Laplacians with are well-studied variational elliptic nonlinear equations, is a limiting case with a convex but not strictly convex functional, and the case admits a variational formulation with a functional that is not convex. In this paper, we augment the equation for the -Laplacian with Cauchy data at the domain’s boundary, which results in a formally overdetermined, nonlinear hyperbolic equation.
This paper presents existence, uniqueness, and stability results for the Cauchy problem of the -Laplacian. In general, the diffusion coefficient can be stably reconstructed only on a subset of described as the domain of influence of the space-like part of the boundary for an appropriate Lorentzian metric. Global reconstructions for specific geometries or based on the construction of appropriate complex geometric optics solutions are also analyzed.
We improve on several weighted inequalities of recent interest by replacing a part of the bounds by weaker estimates involving Wilson’s constant
In particular, we show the following improvement of the first author’s theorem for Calderón–Zygmund operators :
Corresponding type results are obtained from a new extrapolation theorem with appropriate mixed - bounds. This uses new two-weight estimates for the maximal function, which improve on Buckley’s classical bound.
We also derive mixed - type results of Lerner, Ombrosi and Pérez (2009) of the form
An estimate dual to the last one is also found, as well as new bounds for commutators of singular integrals.
A bounded measurable set , of Lebesgue measure 1, in the real line is called spectral if there is a set of real numbers (“frequencies”) such that the exponential functions , , form a complete orthonormal system of . Such a set is called a spectrum of . In this note we prove that any spectrum of a bounded measurable set must be periodic.
We consider finite-energy equivariant solutions for the wave map problem from to which are close to the soliton family. We prove asymptotic orbital stability for a codimension-two class of initial data which is small with respect to a stronger topology than the energy.
In this paper, we consider a discrete restriction associated with KdV equations. Some new Strichartz estimates are obtained. We also establish the local well-posedness for the periodic generalized Korteweg–de Vries equation with nonlinear term provided and the initial data with .
The classical Stein–Tomas restriction theorem is equivalent to the fact that the spectral measure of the square root of the Laplacian on is bounded from to for , where is the conjugate exponent to , with operator norm scaling as . We prove a geometric, or variable coefficient, generalization in which the Laplacian on is replaced by the Laplacian, plus a suitable potential, on a nontrapping asymptotically conic manifold. It is closely related to Sogge’s discrete restriction theorem, which is an estimate on the operator norm of the spectral projection for a spectral window of fixed length. From this, we deduce spectral multiplier estimates for these operators, including Bochner–Riesz summability results, which are sharp for in the range above.
The paper divides naturally into two parts. In the first part, we show at an abstract level that restriction estimates imply spectral multiplier estimates, and are implied by certain pointwise bounds on the Schwartz kernel of -derivatives of the spectral measure. In the second part, we prove such pointwise estimates for the spectral measure of the square root of Laplace-type operators on asymptotically conic manifolds. These are valid for all if the asymptotically conic manifold is nontrapping, and for small in general. We also observe that Sogge’s estimate on spectral projections is valid for any complete manifold with bounded geometry, and in particular for asymptotically conic manifolds (trapping or not), while by contrast, the operator norm on may blow up exponentially as when trapping is present.
In this paper we study periodic homogenization problems for solutions of fully nonlinear PDEs in half-spaces with oscillatory Neumann boundary data. We show the existence and uniqueness of the homogenized Neumann data for a given half-space. Moreover, we show that there exists a continuous extension of the homogenized slope as the normal of the half-space varies over “irrational” directions.
We consider the incompressible Navier–Stokes equations in a two-dimensional exterior domain , with no-slip boundary conditions. Our initial data are of the form , where is the Oseen vortex with unit circulation at infinity and is a solenoidal perturbation belonging to for some . If is sufficiently small, we show that the solution behaves asymptotically in time like the self-similar Oseen vortex with circulation . This is a global stability result, in the sense that the perturbation can be arbitrarily large, and our smallness assumption on the circulation is independent of the domain .
The aim of this note is to show that Alexandrov solutions of the Monge–Ampère equation, with right-hand side bounded away from zero and infinity, converge strongly in if their right-hand sides converge strongly in . As a corollary, we deduce strong stability of optimal transport maps.