Search results for "Mathematica"
showing 10 items of 7971 documents
The fractional Calderón problem: Low regularity and stability
2017
The Calder\'on problem for the fractional Schr\"odinger equation was introduced in the work \cite{GSU}, which gave a global uniqueness result also in the partial data case. This article improves this result in two ways. First, we prove a quantitative uniqueness result showing that this inverse problem enjoys logarithmic stability under suitable a priori bounds. Second, we show that the results are valid for potentials in scale-invariant $L^p$ or negative order Sobolev spaces. A key point is a quantitative approximation property for solutions of fractional equations, obtained by combining a careful propagation of smallness analysis for the Caffarelli-Silvestre extension and a duality argumen…
A sharp stability estimate for tensor tomography in non-positive curvature
2021
Funder: University of Cambridge
Stationary sets of the mean curvature flow with a forcing term
2020
We consider the flat flow approach for the mean curvature equation with forcing in an Euclidean space $\mathbb R^n$ of dimension at least 2. Our main results states that tangential balls in $\mathbb R^n$ under any flat flow with a bounded forcing term will experience fattening, which generalizes the result by Fusco, Julin and Morini from the planar case to higher dimensions. Then, as in the planar case, we are able to characterize stationary sets in $\mathbb R^n$ for a constant forcing term as finite unions of equisized balls with mutually positive distance.
Volume preserving mean curvature flows near strictly stable sets in flat torus
2021
In this paper we establish a new stability result for the smooth volume preserving mean curvature flow in flat torus $\mathbb T^n$ in low dimensions $n=3,4$. The result says roughly that if the initial set is near to a strictly stable set in $\mathbb T^n$ in $H^3$-sense, then the corresponding flow has infinite lifetime and converges exponentially fast to a translate of the strictly stable (critical) set in $W^{2,5}$-sense.
C1,α-regularity for variational problems in the Heisenberg group
2017
We study the regularity of minima of scalar variational integrals of $p$-growth, $1<p<\infty$, in the Heisenberg group and prove the H\"older continuity of horizontal gradient of minima.
Nonlinear Liouville Problems in a Quarter Plane
2016
We answer affirmatively the open problem proposed by Cabr\'e and Tan in their paper "Positive solutions of nonlinear problems involving the square root of the Laplacian" (see Adv. Math. {\bf 224} (2010), no. 5, 2052-2093).
Multi-marginal entropy-transport with repulsive cost
2020
In this paper we study theoretical properties of the entropy-transport functional with repulsive cost functions. We provide sufficient conditions for the existence of a minimizer in a class of metric spaces and prove the $\Gamma$-convergence of the entropy-transport functional to a multi-marginal optimal transport problem with a repulsive cost. We also prove the entropy-regularized version of the Kantorovich duality.
Whitney forms and their extensions
2021
Whitney forms are widely known as finite elements for differential forms. Whitney’s original definition yields first order functions on simplicial complexes, and a lot of research has been devoted to extending the definition to nonsimplicial cells and higher order functions. As a result, the term Whitney forms has become somewhat ambiguous in the literature. Our aim here is to clarify the concept of Whitney forms and explicitly explain their key properties. We discuss Whitney’s initial definition with more depth than usually, giving three equivalent ways to define Whitney forms. We give a comprehensive exposition of their main properties, including the proofs. Understanding of these propert…
Increasing stability in the linearized inverse Schrödinger potential problem with power type nonlinearities
2022
We consider increasing stability in the inverse Schr\"{o}dinger potential problem with power type nonlinearities at a large wavenumber. Two linearization approaches, with respect to small boundary data and small potential function, are proposed and their performance on the inverse Schr\"{o}dinger potential problem is investigated. It can be observed that higher order linearization for small boundary data can provide an increasing stability for an arbitrary power type nonlinearity term if the wavenumber is chosen large. Meanwhile, linearization with respect to the potential function leads to increasing stability for a quadratic nonlinearity term, which highlights the advantage of nonlinearit…
On some partial data Calderón type problems with mixed boundary conditions
2021
In this article we consider the simultaneous recovery of bulk and boundary potentials in (degenerate) elliptic equations modelling (degenerate) conducting media with inaccessible boundaries. This connects local and nonlocal Calderón type problems. We prove two main results on these type of problems: On the one hand, we derive simultaneous bulk and boundary Runge approximation results. Building on these, we deduce uniqueness for localized bulk and boundary potentials. On the other hand, we construct a family of CGO solutions associated with the corresponding equations. These allow us to deduce uniqueness results for arbitrary bounded, not necessarily localized bulk and boundary potentials. T…