We prove a Filippov type existence theorem for solutions of a fractional differential inclusion defined by a nonconvex set-valued map with Dirichlet boundary conditions. The method consists in application of the contraction principle in the space of selections of the set-valued map instead of the space of solutions.

We show that for any K -quasiconformal map of the upper half plane to itself and any ε > 0 , there is a (K + ε) -quasiconformal map of the half plane with the same boundary values which is also biLipschitz with respect to the hyperbolic metric.

We consider the inverse problem to identify an anisotropic conductivity from the Dirichlet-to-Neumann (DtN) map. We first find an explicit reconstruction of the boundary value of less regular anisotropic (transversally isotropic) conductivities and their derivatives. Based on the reconstruction formula, we prove Hölder stability, up to isometry, of the inverse problem using a local DtN map.

Human neuroimaging experiments typically localize motion-selective cortex (MT+) by contrasting responses to stationary and moving stimuli. It has long been suspected that MT+, located on the lateral surface at the temporal-occipital (TO) boundary, contains several distinct visual field maps, although only one coarse map has been measured. Using a novel functional MRI model-based method we ident...

Let Ω ⊂ R, n ≥ 3, be a smooth bounded domain and consider a coupled system in Ω consisting of a conductivity equation ∇ · γ(x)∇u(t, x) = 0 and an anisotropic heat equation κ(x)∂tψ(t, x) = ∇ · (A(x)∇ψ(t, x)) + (γ∇u(t, x)) · ∇u(t, x), t ≥ 0. It is shown that the coefficients γ, κ and A = (ajk) are uniquely determined from the knowledge of the boundary map u|∂Ω 7→ ν ·A∇ψ|∂Ω, where ν is the unit ou...

How close is the Dirichlet-to-Neumann (DtN) map to square root of corresponding boundary Laplacian? This question has been actively investigated in recent years. Somewhat surprisingly, a lot techniques involved can be traced back newly rediscovered manuscript Hörmander from 1950s. We present Hörmander’s approach and its applications, with an emphasis on eigenvalue estimates spectral asymptotics...

We show that the full symbol of the Dirichlet to Neumann map of the k-form Laplace’s equation on a Riemannian manifold (of dimension greater than 2) with boundary determines the full Taylor series of the metric at the boundary. This extends the result of Lee and Uhlmann for the case k = 0. The proof avoids the computation of the full symbol by using the calculus of pseudo-differential operators...

We consider an inverse boundary value problem for the Maxwell equations with boundary data assumed known only in accessible part Γ of the boundary. We aim to prove a uniqueness result using the Dirichlet to Neumann data with measurements limited to an open part of the boundary and we seek to reconstruct the complex refractive index n in the interior of a body. Further, using the impedance map r...

We demonstrate that solving the classical problems mentioned in the title on quadrature domains when the given boundary data is rational is as simple as the method of partial fractions. A by-product of our considerations will be a simple proof that the Dirichlet-to-Neumann map on a double quadrature domain sends rational functions on the boundary to rational functions on the boundary. The resul...