arXiv subjects

Philipp Zimmermann

Publications and source records attributed to Philipp Zimmermann.

28 records · Page 2Linked to original sources

arXiv2022

Stability estimates for the inverse fractional conductivity problem

We study the stability of an inverse problem for the fractional conductivity equation on bounded smooth domains. We obtain a logarithmic stability estimate for the inverse problem under suitable a priori bounds on the globally defined conductivities. The argument has three main ingredients: 1. the logarithmic stability of the related inverse problem for the fractional Schrödinger equation by Rüland and Salo; 2. the Lipschitz stability of the exterior determination problem; 3. utilizing and identifying nonlocal analogies of Alessandrini's work on the stability of the classical Calderón problem. The main contribution of the article is the resolution of the technical difficulties related to the last mentioned step. Furthermore, we show the optimality of the logarithmic stability estimates, following the earlier works by Mandache on the instability of the inverse conductivity problem, and by Rüland and Salo on the analogous problem for the fractional Schrödinger equation.

math.AP↗

arXiv2022

Counterexamples to uniqueness in the inverse fractional conductivity problem with partial data

We construct counterexamples for the partial data inverse problem for the fractional conductivity equation in all dimensions on general bounded open sets. In particular, we show that for any bounded domain $Ω\subset \mathbb{R}^n$ and any disjoint open sets $W_1,W_2 \Subset \mathbb{R}^n \setminus \barΩ$ there always exist two positive, bounded, smooth, conductivities $γ_1,γ_2$, $γ_1 \neq γ_2$, with equal partial exterior Dirichlet-to-Neumann maps $Λ_{γ_1}f|_{W_2} = Λ_{γ_2}f|_{W_2}$ for all $f \in C_c^\infty(W_1)$. The proof uses the characterization of equal exterior data from another work of the authors in combination with the maximum principle of fractional Laplacians. The main technical difficulty arises from the requirement that the conductivities should be strictly positive and have a special regularity property $γ_i^{1/2}-1 \in H^{2s,\frac{n}{2s}}(\mathbb{R}^n)$ for $i=1,2$. We also provide counterexamples on domains that are bounded in one direction when $n \geq 4$ or $s \in (0,n/4]$ when $n=2,3$ using a modification of the argument on bounded domains.

math.AP↗

arXiv2022

Low regularity theory for the inverse fractional conductivity problem

We characterize partial data uniqueness for the inverse fractional conductivity problem with $H^{s,n/s}$ regularity assumptions in all dimensions. This extends the earlier results for $H^{2s,\frac{n}{2s}}\cap H^s$ conductivities by Covi and the authors. We construct counterexamples to uniqueness on domains bounded in one direction whenever measurements are performed in disjoint open sets having positive distance to the domain. In particular, we provide counterexamples in the special cases $s \in (n/4,1)$, $n=2,3$, missing in the literature due to the earlier regularity conditions. We also give a new proof of the uniqueness result which is not based on the Runge approximation property. Our work can be seen as a fractional counterpart of Haberman's uniqueness theorem for the classical Calderón problem with $W^{1,n}$ conductivities when $n=3,4$. One motivation of this work is Brown's conjecture that uniqueness for the classical Calderón problem holds for $W^{1,n}$ conductivities also in dimensions $n \geq 5$.

math.AP↗

arXiv2022

The fractional $p\,$-biharmonic systems: optimal Poincaré constants, unique continuation and inverse problems

This article investigates nonlocal, fully nonlinear generalizations of the classical biharmonic operator $(-Δ)^2$. These fractional $p$-biharmonic operators appear naturally in the variational characterization of the optimal fractional Poincaré constants in Bessel potential spaces. We study the following basic questions for anisotropic fractional $p$-biharmonic systems: existence and uniqueness of weak solutions to the associated interior source and exterior value problems, unique continuation properties (UCP), monotonicity relations, and inverse problems for the exterior Dirichlet-to-Neumann maps. Furthermore, we show the UCP for the fractional Laplacian in all Bessel potential spaces $H^{t,p}$ for any $t\in \mathbb{R}$, $1 \leq p < \infty$ and $s \in \mathbb{R}_+ \setminus \mathbb{N}$: If $u\in H^{t,p}(\mathbb{R}^n)$ satisfies $(-Δ)^su=u=0$ in a nonempty open set $V$, then $u\equiv 0$ in $\mathbb{R}^n$. This property of the fractional Laplacian is then used to obtain a UCP for the fractional $p$-biharmonic systems and plays a central role in the analysis of the associated inverse problems. Our proofs use variational methods and the Caffarelli-Silvestre extension.

math.AP↗

arXiv2022

The global inverse fractional conductivity problem

We prove \emph{global} uniqueness for an inverse problem for the fractional conductivity equation on domains that are bounded in one direction. The conductivities are assumed to be isotropic and nontrivial in the exterior of the domain, while the data is given in the form of partial Dirichlet-to-Neumann (DN) maps measured in nondisjoint open subsets of the exterior. This can be seen as the fractional counterpart of the classical inverse conductivity problem. The proof is based on a unique continuation property (UCP) for the DN maps and an exterior determination method from the partial exterior DN maps. This is analogous to the classical boundary determination method by Kohn and Vogelius. The most important technical novelty is the construction of sequences of special solutions to the fractional conductivity equation whose Dirichlet energies in the limit can be concentrated at any given point in the exterior. This is achieved independently of the UCP and despite the nonlocality of the equation. Due to the recent counterexamples by the last two authors, our results almost completely characterize uniqueness for the inverse fractional conductivity problem with partial data for isotropic global conductivities.

math.AP↗

arXiv2022

Fractional Calderón problems and Poincaré inequalities on unbounded domains

We generalize many recent uniqueness results on the fractional Calderón problem to cover the cases of all domains with nonempty exterior. The highlight of our work is the characterization of uniqueness and nonuniqueness of partial data inverse problems for the fractional conductivity equation on domains that are bounded in one direction for conductivities supported in the whole Euclidean space and decaying to a constant background conductivity at infinity. We generalize the uniqueness proof for the fractional Calderón problem by Ghosh, Salo and Uhlmann to a general abstract setting in order to use the full strength of their argument. This allows us to observe that there are also uniqueness results for many inverse problems for higher order local perturbations of a lower order fractional Laplacian. We give concrete example models to illustrate these curious situations and prove Poincaré inequalities for the fractional Laplacians of any order on domains that are bounded in one direction. We establish Runge approximation results in these general settings, improve regularity assumptions also in the cases of bounded sets and prove general exterior determination results. Counterexamples to uniqueness in the inverse fractional conductivity problem with partial data are constructed in another companion work.

math.AP↗

arXiv2020

Photo-physics and electronic structure of lateral graphene/MoS2 and metal/MoS2 junctions

Integration of semiconducting transition metal dichalcogenides (TMDs) into functional optoelectronic circuitries requires an understanding of the charge transfer across the interface between the TMD and the contacting material. Here, we use spatially resolved photocurrent microscopy to demonstrate electronic uniformity at the epitaxial graphene/molybdenum disulfide (EG/MoS2) interface. A 10x larger photocurrent is extracted at the EG/MoS2 interface when compared to metal (Ti/Au) /MoS2 interface. This is supported by semi-local density-functional theory (DFT), which predicts the Schottky barrier at the EG/MoS2 interface to be ~2x lower than Ti/MoS2. We provide a direct visualization of a 2D material Schottky barrier through combination of angle resolved photoemission spectroscopy with spatial resolution selected to be ~300 nm (nano-ARPES) and DFT calculations. A bending of ~500 meV over a length scale of ~2-3 micrometer in the valence band maximum of MoS2 is observed via nano-ARPES. We explicate a correlation between experimental demonstration and theoretical predictions of barriers at graphene/TMD interfaces. Spatially resolved photocurrent mapping allows for directly visualizing the uniformity of built-in electric fields at heterostructure interfaces, providing a guide for microscopic engineering of charge transport across heterointerfaces. This simple probe-based technique also speaks directly to the 2D synthesis community to elucidate electronic uniformity at domain boundaries alongside morphological uniformity over large areas.

physics.app-ph↗

arXiv2019

Space-time error estimates for deep neural network approximations for differential equations

Over the last few years deep artificial neural networks (DNNs) have very successfully been used in numerical simulations for a wide variety of computational problems including computer vision, image classification, speech recognition, natural language processing, as well as computational advertisement. In addition, it has recently been proposed to approximate solutions of partial differential equations (PDEs) by means of stochastic learning problems involving DNNs. There are now also a few rigorous mathematical results in the scientific literature which provide error estimates for such deep learning based approximation methods for PDEs. All of these articles provide spatial error estimates for neural network approximations for PDEs but do not provide error estimates for the entire space-time error for the considered neural network approximations. It is the subject of the main result of this article to provide space-time error estimates for DNN approximations of Euler approximations of certain perturbed differential equations. Our proof of this result is based (i) on a certain artificial neural network (ANN) calculus and (ii) on ANN approximation results for products of the form $[0,T]\times \mathbb{R}^d\ni (t,x)\mapsto tx\in \mathbb{R}^d$ where $T\in (0,\infty)$, $d\in \mathbb{N}$, which we both develop within this article.

math.NA↗

arXiv2017

Towards femtosecond on-chip electronics

To combine the advantages of ultrafast femtosecond optics with an on-chip communcation scheme, optical signals with a frequency of several hundreds of THz need to be down-converted to coherent electronic signals of GHz or less. So far, this has not been achieved because of the impedance mismatch within electronic circuits and their overall slow response-time. Here, we demonstrate that 14 fs optical pulses in the near-infrared (NIR) can drive electronic on-chip circuits with a bandwidth up to 10 THz. The corresponding electronic pulses propagate in microscopic striplines on a millimeter scale. We exploit femtosecond photoswitches based on tunneling barriers in nanoscale metal junctions to drive the pulses. The non-linear ultrafast response is based on a combination of plasmonically enhanced, multi-photon absorption and quantum tunneling, and gives rise to a field emission of ballistic electrons propagating across the nanoscale junctions. Our results pave the way towards femtosecond electronics integrated in waferscale quantum circuits.

cond-mat.mes-hall↗

arXiv2016

Switchable topological phonon channels

Guiding energy deliberately is one of the central elements in engineering and information processing. It is often achieved by designing specific transport channels in a suitable material. Topological metamaterials offer a way to construct stable and efficient channels of unprecedented versatility. However, due to their stability it can be tricky to terminate them or to temporarily shut them off without changing the material properties massively. While a lot of effort was put into realizing mechanical topological metamaterials, almost no works deal with manipulating their edge channels in sight of applications. Here, we take a step in this direction, by taking advantage of local symmetry breaking potentials to build a switchable topological phonon channel.

cond-mat.mes-hall↗