arXiv subjects
Publications and source records attributed to Lukas Koch.
We consider an iterative procedure to solve quasilinear elliptic systems with $p$-growth. The scheme was first considered by Koshelev in the quadratic case $p=2$. We present numerical applications as well as applications to higher regularity properties.
We prove global $W^{1,q}(Ω,\mathbb{R}^N)$-regularity for minimisers of $\mathscr{F}(u)=\int_ΩF(x,\mathrm{D}u)\mathrm{d} x$ satisfying $u\geq ψ$ for a given Sobolev obstacle $ψ$. $W^{1,q}(Ω,\mathbb{R}^m)$ regularity is also proven for minimisers of the associated relaxed functional. Our main assumptions on $F(x,z)$ are a uniform $α$-Hölder continuity assumption in $x$ and natural $(p,q)$-growth conditions in $z$ with $q<\frac{(n+α)p}{n}$. In the autonomous case $F\equiv F(z)$ we can improve the gap to $q<\frac{np}{n-1}$, a result new even in the unconstrained case.
Not all experiments publish their results with a description of the correlations between the data points. This makes it difficult to do hypothesis tests or model fits with that data, since just assuming no correlation can lead to an over- or underestimation of the resulting uncertainties. This work presents robust test statistics that can be used with data sets with missing correlation information. They are exact in the case of no correlation and either guaranteed to be conservative -- i.e. the uncertainty is never underestimated -- in the presence of correlations, or they are also exact in the degenerate case of perfect correlation between the data points.
We study the symmetry and uniqueness of maps which minimise the $np$-Dirichlet energy, under the constraint that their Jacobian is a given radially symmetric function $f$. We find a condition on $f$ which ensures that the minimisers are symmetric and unique. In the absence of this condition we construct an explicit $f$ for which there are uncountably many distinct energy minimisers, none of which is symmetric. Even if we prescribe the maps to be the identity on the boundary of a ball we show that the minimisers need not be symmetric. This gives a negative answer to a question of Hélein (Ann. Inst. H. Poincaré Anal. Non Linéaire 11 (1994), no. 3, 275-296).
We prove global $W^{1,q}(Ω,\mathbb{R}^m)$-regularity for minimisers of convex functionals of the form $\mathscr{F}(u)=\int_ΩF(x,Du)\mathrm{d} x$. $W^{1,q}(Ω,\mathbb{R}^m)$ regularity is also proven for minimisers of the associated relaxed functional. Our main assumptions on $F(x,z)$ are a uniform $α$-Hölder continuity assumption in $x$ and controlled $(p,q)$-growth conditions in $z$ with $q<\frac{(n+α)p}{n}$.
The exact way of treating flux shape uncertainties in unfolded, flux-averaged neutrino cross-section measurements can lead to subtle issues when comparing the results to model predictions. There is a difference between reporting a cross section in the (unknown) real flux, and reporting a cross section that was extrapolated from the (unknown) real flux to a fixed reference flux. A lot of (most?) current analyses do the former, while the results are compared to model predictions as if they were the latter. This leads to (part of) the flux shape uncertainty being ignored, potentially leading to wrong physics conclusions. The size of the effect is estimated to be sub-dominant, but non-negligible in two example measurements from T2K and MINERvA. This paper describes how the issue arises and provides instructions for possible ways how to treat the flux shape uncertainties correctly.
For a nonlinear operator $T$ satisfying certain structural assumptions, our main theorem states that the following claims are equivalent: i) $T$ is surjective, ii) $T$ is open at zero, and iii) $T$ has a bounded right inverse. The theorem applies to numerous scale-invariant PDEs in regularity regimes where the equations are stable under weak$^*$ convergence. Two particular examples we explore are the Jacobian equation and the equations of incompressible fluid flow. For the Jacobian, it is a long standing open problem to decide whether it is onto between the critical Sobolev space and the Hardy space. Towards a negative answer, we show that, if the Jacobian is onto, then it suffices to rule out the existence of surprisingly well-behaved solutions. For the incompressible Euler equations, we show that, for any $p<\infty$, the set of initial data for which there are dissipative weak solutions in $L^p_t L^2_x$ is meagre in the space of solenoidal $L^2$ fields. Similar results hold for other equations of incompressible fluid dynamics.
We study existence and regularity of solutions to the Dirichlet problem for the prescribed Jacobian equation, $\det Du = f$, where $f$ is integrable and bounded away from zero. In particular, we take $f\in L^p$, where $p > 1$, or in $L\log L$. We prove that for a Baire-generic $f$ in either space there are no solutions with the expected regularity.
We prove partial and full boundary regularity for manifold constrained $p(x)$-harmonic maps.
Many emerging applications in microscale engineering rely on the fabrication of three-dimensional architectures in inorganic materials. Small-scale additive manufacturing (AM) aspires to provide flexible and facile access to these geometries. Yet, the synthesis of device-grade inorganic materials is still a key challenge towards the implementation of AM in microfabrication. Here, we present a comprehensive overview of the microstructural and mechanical properties of metals fabricated by most state-of-the-art AM methods that offer a spatial resolution $\leq$10$μ$m. Standardized sets of samples were studied by cross-sectional electron microscopy, nanoindentation and microcompression. We show that current microscale AM techniques synthesize metals with a wide range of microstructures and elastic and plastic properties, including materials of dense and crystalline microstructure with excellent mechanical properties that compare well to those of thin-film nanocrystalline materials. The large variation in materials performance can be related to the individual microstructure, which in turn is coupled to the various physico-chemical principles exploited by the different printing methods. The study provides practical guidelines for users of small-scale additive methods and establishes a baseline for the future optimization of the properties of printed metallic objects $-$ a significant step towards the potential establishment of AM techniques in microfabrication.
The current canonical approach to publishing cross-section data is to unfold the reconstructed distributions. Detector effects like efficiency and smearing are undone mathematically, yielding distributions in true event properties. This is an ill-posed problem, as even small statistical variations in the reconstructed data can lead to large changes in the unfolded spectra. This work presents an alternative or complementary approach: the response-matrix-centred forward-folding approach. It offers a convenient way to forward-fold model expectations in truth space to reconstructed quantities. These can then be compared to the data directly, similar to what is usually done with full detector simulations within the experimental collaborations. For this, the detector response (efficiency and smearing) is parametrised as a matrix. The effects of the detector on the measurement of a given model is simulated by simply multiplying the binned truth expectation values by this response matrix. Systematic uncertainties in the detector response are handled by providing a set of matrices according to the prior distribution of the detector properties and marginalising over them. Background events can be included in the likelihood calculation by giving background events their own bins in truth space. To facilitate a straight-forward use of response matrices, a new software framework has been developed: the Response Matrix Utilities (ReMU). ReMU is a Python package distributed via the Python Package Index. It only uses widely available, standard scientific Python libraries and does not depend on any custom experiment-specific software. It offers all methods needed to build response matrices from Monte Carlo data sets, use the response matrix to forward-fold truth-level model predictions, and compare the predictions to real data using Bayesian or frequentist statistical inference.
The T2K near-detector, ND280, employs three large argon gas TPCs (Time Projection Chambers) for particle tracking and identification. The gas inside the TPCs can be used as an active target to study the neutrino interactions in great detail. The low density of the gas leads to very low track energy thresholds, allowing the reconstruction of very low momentum tracks, e.g. protons with kinetic energies down to $\mathcal{O}$(1 MeV). Since different nuclear interaction models vary considerably in their predictions of those low momentum track multiplicities, this makes neutrino interactions on gases a powerful probe to test those models. The TPCs operate with an argon-based gas mixture (95% by volume) and have been exposed to the T2K neutrino beam since the beginning of the experiment in 2010. Due to the low total mass of the gas, neutrino argon interactions happen only rarely, compared to the surrounding scintillator-based detectors. We expect about 600 such events in the recorded data so far (about 200 in the fiducial volume). We are able to separate those events from the background and thus demonstrate the viability of using gaseous argon as a target for a neutrino beam. This enables us to do a cross- section measurement on gaseous argon, the first measurement of this kind. All previous neutrino cross-section measurements on argon were performed in liquid argon TPCs.