Boundary Tracing Research Articles

Stabilisation of linear waves with inhomogeneous Neumann boundary conditions

ABSTRACT We study linear damped and viscoelastic wave equations evolving on a bounded domain. For both models, we assume that waves are subject to an inhomogeneous Neumann boundary condition on a portion of the domain's boundary. The analysis of these models presents additional interesting features and challenges compared to their homogeneous counterparts. In the present context, energy depends on the boundary trace of velocity. It is not clear in advance how this quantity should be controlled based on the given data, due to regularity issues. However, we establish global existence and also prove uniform stabilisation of solutions with decay rates characterised by the Neumann input. We supplement these results with numerical simulations in which the data do not necessarily satisfy the given assumptions for decay. These simulations provide, at a numerical level, insights into how energy could possibly change in the presence of, for example, improper data.

Sharp Trace and Korn Inequalities for Differential Operators

AbstractWe establish sharp trace and Korn-type inequalities that involve vectorial differential operators, the focus being on situations where global singular integral estimates are not available. Starting from a novel approach to sharp Besov boundary traces by Riesz potentials and oscillations that equally applies to $$p=1$$ p = 1 , a case difficult to be handled by harmonic analysis techniques, we then classify boundary trace- and Korn-type inequalities. For $$p=1$$ p = 1 and so despite the failure of the Calderón-Zygmund theory, we prove that sharp trace estimates can be systematically reduced to full k-th order gradient estimates. Moreover, for $$1<p<\infty $$ 1 < p < ∞ , where sharp trace estimates yield Korn-type inequalities on smooth domains, we show for the basically optimal class of John domains that Korn-type inequalities persist – even though the reduction to global Calderón-Zygmund estimates by extension operators might not be possible.

Boundary Traces of Holomorphic Functions on the Unit Ball in $$\mathbb {C}^n$$

Boundary Traces of Holomorphic Functions on the Unit Ball in $$\mathbb {C}^n$$

In-situ analysis of SCC initiation on Alloy 600 surface in primary water environment using digital image correlation with electron backscatter diffraction

This study examined the stress corrosion cracking (SCC) phenomenon of Alloy 600 in a high-temperature primary water environment. The investigation involved the observation of local plastic deformation and SCC initiation on the specimen surface using an observation apparatus, comprising a laser confocal microscope (LCM) and an autoclave equipped with a diamond window view-port. Prior to conducting the SCC initiation test, the crystalline structure of the specimen, including grain boundary (GB) characteristics, Schmid factor, and trace inclination angle, underwent examination using an electron backscatter diffraction (EBSD) technique. The local strain distribution was subsequently analyzed using a digital image correlation (DIC) method, employing LCM images obtained during the SCC experiments. The ex-situ EBSD analysis indicated that SCC mainly initiated at random high-angle grain boundaries (HAGB) with trace inclination angles close to 90° relative to the tensile axis. DIC analysis further revealed that most GBs in the vicinity of the highly strained area cracked among random HAGBs with a high trace inclination angle to the tensile axis. This confirms that the local strain gradient at GBs plays an important role in SCC initiation, serving as a more suitable indicator of an additional stress component to effective normal stress acting on GBs compared to the Schmid factor mismatch.

Trace Operator on von Koch’s Snowflake

AbstractWe study properties of the boundary trace operator on the Sobolev space $$W^1_1(\Omega )$$ W 1 1 ( Ω ) . Using the density result by Koskela and Zhang (Arch. Ration. Mech. Anal. 222(1), 1-14 2016), we define a surjective operator $$Tr: W^1_1(\Omega _K)\rightarrow X(\Omega _K)$$ T r : W 1 1 ( Ω K ) → X ( Ω K ) , where $$\Omega _K$$ Ω K is von Koch’s snowflake and $$X(\Omega _K)$$ X ( Ω K ) is a trace space with the quotient norm. Since $$\Omega _K$$ Ω K is a uniform domain whose boundary is Ahlfors-regular with an exponent strictly bigger than one, it was shown by L. Malý (2017) that there exists a right inverse to Tr, i.e. a linear operator $$S: X(\Omega _K) \rightarrow W^1_1(\Omega _K)$$ S : X ( Ω K ) → W 1 1 ( Ω K ) such that $$Tr \circ S= Id_{X(\Omega _K)}$$ T r ∘ S = I d X ( Ω K ) . In this paper we provide a different, purely combinatorial proof based on geometrical structure of von Koch’s snowflake. Moreover we identify the isomorphism class of the trace space as $$\ell _1$$ ℓ 1 . As an additional consequence of our approach we obtain a simple proof of the Peetre’s theorem (Special Issue 2, 277-282 1979) about non-existence of the right inverse for domain $$\Omega $$ Ω with regular boundary, which explains Banach space geometry cause for this phenomenon.

Semilinear elliptic equations involving power nonlinearities and Hardy potentials with boundary singularities

Let $\Omega \subset \mathbb {R}^N$ ( $N\geq 3$ ) be a $C^2$ bounded domain and $\Sigma \subset \partial \Omega$ be a $C^2$ compact submanifold without boundary, of dimension $k$ , $0\leq k \leq N-1$ . We assume that $\Sigma = \{0\}$ if $k = 0$ and $\Sigma =\partial \Omega$ if $k=N-1$ . Let $d_{\Sigma }(x)=\mathrm {dist}\,(x,\Sigma )$ and $L_\mu = \Delta + \mu \,d_{\Sigma }^{-2}$ , where $\mu \in {\mathbb {R}}$ . We study boundary value problems ( $P_\pm$ ) $-{L_\mu} u \pm |u|^{p-1}u = 0$ in $\Omega$ and $\mathrm {tr}_{\mu,\Sigma}(u)=\nu$ on $\partial \Omega$ , where $p>1$ , $\nu$ is a given measure on $\partial \Omega$ and $\mathrm {tr}_{\mu,\Sigma}(u)$ denotes the boundary trace of $u$ associated to $L_\mu$ . Different critical exponents for the existence of a solution to ( $P_\pm$ ) appear according to concentration of $\nu$ . The solvability for problem ( $P_+$ ) was proved in [3, 29] in subcritical ranges for $p$ , namely for $p$ smaller than one of the critical exponents. In this paper, assuming the positivity of the first eigenvalue of $-L_\mu$ , we provide conditions on $\nu$ expressed in terms of capacities for the existence of a (unique) solution to ( $P_+$ ) in supercritical ranges for $p$ , i.e. for $p$ equal or bigger than one of the critical exponents. We also establish various equivalent criteria for the existence of a solution to ( $P_-$ ) under a smallness assumption on $\nu$ .

Enhanced and efficient scaled boundary finite elements for mechanical problems under the action of body loads

The Scaled Boundary Finite Element Method (SBFEM) can be viewed as a semi-analytical approach, which has been successfully applied to singular mechanical problems with vanishing force terms. Interpreted in the context of composite Duffy approximations based on star-shape polytopes (S-elements), and using an operator-adapted principle, the shape functions are determined by imposing an energy-orthogonality constraint. In the presence of non-vanishing forcing terms, the SBFEM approximation converges at sub-optimal rates. In the current paper, a strategy is proposed to overcome this drawback by the enrichment of the classic SBFEM spaces with energy-orthogonal bubble Duffy functions. The orthogonality property leads to a decoupling of the approximate solutions into components of boundary trace and interior (bubble) functions. The bubble functions can be solved separately, thus reducing the computational cost. A priori energy-error estimates are provided demonstrating optimal rates of convergence. The convergence rates are confirmed by two and three-dimensional numerical verification tests.

Deformation twin traces on gold surfaces: A pathway to tailored epitaxial growth of 1D semiconductors.

The field of one-dimensional semiconducting materials holds a wide variety of captivating applications, such as photovoltaic cells, electronic devices, catalysis cells, lasers, and more. The tunability of electrical, mechanical, or optical attributes of a semiconductor crystal relies on the ability to control and pattern the crystal's growth direction, orientation, and dimensions. In this study, we harvest the unique properties of crystallographic defects in Au substrates, specifically twin boundaries, to fabricate selective epitaxial growth of semiconducting nanocrystals. Different crystallographic defects were previously shown to enhance materials properties, such as, screw dislocations providing spiral crystal growth, dislocation outcrops, and vacancies increasing their catalytic activity, dislocation strengthening, and atomic doping changing the crystal's electrical properties. Here, we present a unique phenomenon of directed growth of semiconductor crystals of gold(I)-cyanide (AuCN) on the surface of thin Au layers, using traces of deformation twins on the surface. We show that emergence of deformation twins to the {111} Au surface leads to the formation of ledges, exposing new {001} and {111} facets on the surface. We propose that this phenomenon leads to epitaxial growth of AuCN on the freshly exposed {111} facets of the twin boundary trace ledges. Specific orientations of the twin boundaries with respect to the Au surface allow for patterned growth of AuCN in the <110> orientations. Nano-scale patterning of AuCN semiconductors may provide an avenue for property tuning, particularly the band gap acquired.

On the symmetric rearrangement of the gradient of a Sobolev function

In this paper, we generalize a classical comparison result for solutions to Hamilton–Jacobi equations with Dirichlet boundary conditions, to solutions to Hamilton–Jacobi equations with non-zero boundary trace. As a consequence, we prove the isoperimetric inequality for the torsional rigidity (with Robin boundary conditions) and for other functionals involving such boundary conditions.

Overview and statistical analysis of boundary layer clouds and precipitation over the western North Atlantic Ocean

Abstract. Due to their fast evolution and large natural variability in macro- and microphysical properties, the accurate representation of boundary layer clouds in current climate models remains a challenge. One of the regions with large intermodel spread in the Coupled Model Intercomparison Project Phase 6 ensemble is the western North Atlantic Ocean. Here, statistically representative in situ measurements can help to develop and constrain the parameterization of clouds in global models. To this end, we performed comprehensive measurements of boundary layer clouds, aerosol, trace gases, and radiation in the western North Atlantic Ocean during the NASA Aerosol Cloud meTeorology Interactions oVer the western ATlantic Experiment (ACTIVATE) mission. In total, 174 research flights with 574 flight hours for cloud and precipitation measurements were performed with the HU-25 Falcon during three winter (February–March 2020, January–April 2021, and November 2021–March 2022) and three summer seasons (August–September 2020, May–June 2021, and May–June 2022). Here we present a statistical evaluation of 16 140 individual cloud events probed by the fast cloud droplet probe and the two-dimensional stereo cloud probe during 155 research flights in a representative and repetitive flight strategy allowing for robust statistical data analyses. We show that the vertical profiles of distributions of the liquid water content and the cloud droplet effective diameter (ED) increase with altitude in the marine boundary layer. Due to higher updraft speeds, higher cloud droplet number concentrations (Nliquid) were measured in winter compared to summer despite lower cloud condensation nucleus abundance. Flight cloud cover derived from statistical analysis of in situ data is reduced in summer and shows large variability. This seasonal contrast in cloud coverage is consistent with a dominance of a synoptic pattern in winter that favors conditions for the formation of stratiform clouds at the western edge of cyclones (post-cyclonic). In contrast, a dominant summer anticyclone is concomitant with the occurrence of shallow cumulus clouds and lower cloud coverage. The evaluation of boundary layer clouds and precipitation in the Nliquid ED phase space sheds light on liquid, mixed-phase, and ice cloud properties and helps to categorize the cloud data. Ice and liquid precipitation, often masked in cloud statistics by a high abundance of liquid clouds, is often observed throughout the cloud. The ACTIVATE in situ cloud measurements provide a wealth of cloud information useful for assessing airborne and satellite remote-sensing products, for global climate and weather model evaluations, and for dedicated process studies that address precipitation and aerosol–cloud interactions.

Heat and Martin Kernel estimates for Schrödinger operators with critical Hardy potentials

Let Ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega $$\\end{document} be a bounded domain in RN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathbb {R}}}^N$$\\end{document} with C2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^2$$\\end{document} boundary and let K⊂∂Ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K\\subset \\partial \\Omega $$\\end{document} be either a C2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^2$$\\end{document} submanifold of the boundary of codimension k<N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k<N$$\\end{document} or a point. In this article we study various problems related to the Schrödinger operator Lμ=-Δ-μdK-2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_{\\mu } =-\\Delta - \\mu d_K^{-2}$$\\end{document} where dK\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d_K$$\\end{document} denotes the distance to K and μ≤k2/4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu \\le k^2/4$$\\end{document}. We establish parabolic boundary Harnack inequalities as well as related two-sided heat kernel and Green function estimates. We construct the associated Martin kernel and prove existence and uniqueness for the corresponding boundary value problem with data given by measures. To prove our results we introduce among other things a suitable notion of boundary trace. This trace is different from the one used by Marcus and Nguyen (Math Ann 374(1–2):361–394, 2019) thus allowing us to cover the whole range μ≤k2/4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu \\le k^2/4$$\\end{document}.

Farmland boundary extraction based on the AttMobile-DeeplabV3+ network and least squares fitting of straight lines.

The rapid extraction of farmland boundaries is key to implementing autonomous operation of agricultural machinery. This study addresses the issue of incomplete farmland boundary segmentation in existing methods, proposing a method for obtaining farmland boundaries based on unmanned aerial vehicle (UAV) remote sensing images. The method is divided into two steps: boundary image acquisition and boundary line fitting. To acquire the boundary image, an improved semantic segmentation network, AttMobile-DeeplabV3+, is designed. Subsequently, a boundary tracing function is used to track the boundaries of the binary image. Lastly, the least squares method is used to obtain the fitted boundary line. The paper validates the method through experiments on both crop-covered and non-crop-covered farmland. Experimental results show that on crop-covered and non-crop-covered farmland, the network's intersection over union (IoU) is 93.25% and 93.14%, respectively; the pixel accuracy (PA) for crop-covered farmland is 96.62%. The average vertical error and average angular error of the extracted boundary line are 0.039 and 1.473°, respectively. This research provides substantial and accurate data support, offering technical assistance for the positioning and path planning of autonomous agricultural machinery.

Minimizing properties of networks via global and local calibrations

AbstractIn this note, we prove that minimal networks enjoy minimizing properties for the length functional. A minimal network is, roughly speaking, a subset of composed of straight segments joining at triple junctions forming angles equal to ; in particular such objects are just critical points of the length functional a priori. We show that a minimal network : (i) minimizes mass among currents with coefficients in an explicit group (independent of ) having the same boundary of , (ii) identifies the interfaces of a partition of a neighborhood of solving the minimal partition problem among partitions with same boundary traces. Consequences and sharpness of such results are discussed. The proofs reduce to rather simple and direct arguments based on the exhibition of (global or local) calibrations associated to the minimal network.

Boundary output feedback stabilization of state delayed reaction–diffusion PDEs

This paper addresses the problem of output feedback stabilization of a class of 1-D reaction–diffusion PDEs in the presence of a state delay in the reaction term. The control input applies through a Robin boundary condition. The system output is selected as either a Dirichlet or Neumann boundary trace. The developed control design procedure does not aim to compensate, but rather to dominate the state delayed term. This is achieved by introducing an observer that only estimates a finite number of modes of the PDE while applying a suitable pole shifting-based procedure on a truncated model that captures the unstable part of the plant. We show that the reported output feedback control strategy always achieves, for an arbitrarily given value of the state delay, the exponential stabilization of the plant in H1-norm. A notable feature of the domination-based approach is that the tuning of the controller and observer gains, as well as the subsequent stability conditions, appears to be independent of the value of the state delay.

Iterative learning algorithms for boundary tracing of fractional diffusion equations

AbstractWe study P‐type and PI ‐type iterative learning control (ILC) schemes for boundary tracing problem of nonhomogeneous fractional diffusion equations. Based on Sobolev imbedding theorem, we derive sufficient conditions for the convergence of four ILC schemes in the sense of ‐norm. Numerical examples are presented to illustrate effectiveness of the proposed control methods. The results show that closed‐loop ILC scheme converges faster than open‐loop ILC scheme; moreover, PI ‐type ILC scheme outperforms P‐type and PI‐type ILC schemes in terms of the convergence speed.

Forgery Detection by Weighted Complementarity between Significant Invariance and Detail Enhancement

Generative adversarial networks have shown impressive results in the modeling of movies and games, but what if such powerful image generation capability is used to harm the Multimedia? The face replacement methods represented by Deepfakes are becoming a threat to everyone, so the development of image authenticity detection methods has become a top priority. For achieving accurate detection resistant to compression effects, we propose a weighted complementary dual-stream detection method. Firstly, to alleviate the influence of image compression on manipulation detection, we propose the concept of pixel-wise saliency invariance. We map fake images onto saliency maps via Quaternary Fourier Transform, which discovers the invariant properties of image phase spectra on different compressions. Meanwhile, to capture boundary traces more easily, we propose the concept of pixel-wise detail enhancement. We apply Bilateral Filtering to preserve the texture edges of fake images and amplify the fake boundaries. Finally, to take full advantage of the two proposed concepts, a weighted complementary dual-stream network (WCD Network) is designed as a classifier to fuse features and identify real and fake. On different benchmarks like FaceForensics++(FF++), Celeb-DF and DFDC, the experimental results show that the proposed method has the average best detection accuracy compared to existing methods.

Inverse source problem with a posteriori boundary measurement for fractional diffusion equations

In this article, we study inverse source problems for time‐fractional diffusion equations from a posteriori boundary measurement. Using the memory effect of these class of equations, we solve these inverse problems for several class of space‐ or time‐dependent source terms. We prove also the unique determination of a general class of space–time‐dependent separated variables source terms from such measurement. Our approach is based on the study of singularities of the Laplace transform in time of boundary traces of solutions of time‐fractional diffusion equations.

Boundary value problems with rough boundary data

We consider linear boundary value problems for higher-order parameter-elliptic equations, where the boundary data do not belong to the classical trace spaces. We employ a class of Sobolev spaces of mixed smoothness that admits a generalized boundary trace with values in Besov spaces of negative order. We prove unique solvability for rough boundary data in the half-space and in sufficiently smooth domains. As an application, we show that the operator related to the linearized Cahn–Hilliard equation with dynamic boundary conditions generates a holomorphic semigroup in Lp(R+n)×Lp(Rn−1).

A Small-Angle Neutron Scattering, Calorimetry and Densitometry Study to Detect Phase Boundaries and Nanoscale Domain Structure in a Binary Lipid Mixture

Techniques that can probe nanometer length scales, such as small-angle neutron scattering (SANS), have become increasingly popular to detect phase separation in membranes. But to extract the phase composition and domain structure from the SANS traces, complementary information is needed. Here, we present a SANS, calorimetry and densitometry study of a mixture of two saturated lipids that exhibits solidus-liquidus phase coexistence: 1,2-dipalmitoyl-d62-sn-glycero-3-phosphocholine (dDPPC, tail-deuterated DPPC) and 1,2-dilauroyl-sn-glycero-3-phosphocholine (DLPC). With calorimetry, we investigated the phase diagram for this system and found that the boundary traces for both multilamellar vesicles (MLVs) as well as 50 nm unilamellar vesicles overlap. Because the solidus boundary was mostly inaccessible by calorimetry, we investigated it by both SANS and molecular volume measurements for a 1:1 dDPPC:DLPC lipid mixture. From the temperature behavior of the molecular volume for the 1:1 dDPPC:DLPC mixture, as well as the individual molecular volume of each lipid species, we inferred that the liquidus phase consists of only fluid-state lipids while the solidus phase consists of lipids that are in gel-like states. Using this solidus-liquidus phase model, the SANS data were analyzed with an unrestricted shape model analysis software: MONSA. The resulting fits show irregular domains with dendrite-like features as those previously observed on giant unilamellar vesicles (GUVs). The surface pair correlation function describes a characteristic domain size for the minority phase that decreases with temperature, a behavior found to be consistent with a concomitant decrease in membrane mismatch between the liquidus and solidus phases.

Asymptotics Near Extinction for Nonlinear Fast Diffusion on a Bounded Domain

On a smooth bounded Euclidean domain, Sobolev-subcritical fast diffusion with vanishing boundary trace is known to lead to finite-time extinction, with a vanishing profile selected by the initial datum. In rescaled variables, we quantify the rate of convergence to this profile uniformly in relative error, showing the rate is either exponentially fast (with a rate constant predicted by the spectral gap), or algebraically slow (which is only possible in the presence of non-integrable zero modes). In the first case, the nonlinear dynamics are well-approximated by exponentially decaying eigenmodes up to at least twice the gap; this refines and confirms a 1980 conjecture of Berryman and Holland. We also improve on a result of Bonforte and Figalli by providing a new and simpler approach which is able to accommodate the presence of zero modes, such as those that occur when the vanishing profile fails to be isolated (and possibly belongs to a continuum of such profiles).