Regular Boundary Research Articles

Spatiotemporal Fusion Prediction of Sea Surface Temperatures Based on the Graph Convolutional Neural and Long Short-Term Memory Networks

Sea surface temperature (SST) prediction plays an important role in scientific research, environmental protection, and other marine-related fields. However, most of the current prediction methods are not effective enough to utilize the spatial correlation of SSTs, which limits the improvement of SST prediction accuracy. Therefore, this paper first explores spatial correlation mining methods, including regular boundary division, convolutional sliding translation, and clustering neural networks. Then, spatial correlation mining through a graph convolutional neural network (GCN) is proposed, which solves the problem of the dependency on regular Euclidian space and the lack of spatial correlation around the boundary of groups for the above three methods. Based on that, this paper combines the spatial advantages of the GCN and the temporal advantages of the long short-term memory network (LSTM) and proposes a spatiotemporal fusion model (GCN-LSTM) for SST prediction. The proposed model can capture SST features in both the spatial and temporal dimensions more effectively and complete the SST prediction by spatiotemporal fusion. The experiments prove that the proposed model greatly improves the prediction accuracy and is an effective model for SST prediction.

Rectilinear Building Footprint Regularization Using Deep Learning

Abstract. Nowadays, deep learning allows to automatically learn features from data. Buildings are one of the most important objects in urban environments. They are used in applications such as inputs to building reconstruction, disaster monitoring, city planing and environment modelling for autonomous driving. However, it is not enough to represent them in raster format, since applications require buildings as polygons. We use an existing, learning based approach to extract building footprints from ortho imagery and digital surface model (DSM) and propose a pipeline for building polygon extraction, which we call primary orientation learning (POL). The first step is to extract initial polygons, that contain a vertex for each pixel in the boundary of the footprint. Afterwards, the two primary orientation angles are regressed continuously. Using these orientation, we insert vertices such that all consecutive edges are perpendicular. To the best of our knowledge, our approach is the first to predict a continuous orientation angle for building boundary regularization. Furthermore, the proposed method is highly efficient with an average processing time of 2.879 ms for a single building.

Regularity of flat free boundaries for two-phase p(x)-Laplacian problems with right hand side

We consider viscosity solutions to two-phase free boundary problems for the p(x)-Laplacian with non-zero right hand side. We prove that flat free boundaries are C1,γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^{1,\\gamma }$$\\end{document}. No assumption on the Lipschitz continuity of solutions is made. These regularity results are the first ones in literature for two-phase free boundary problems for the p(x)-Laplacian and also for two-phase problems for singular/degenerate operators with non-zero right hand side. They are new even when p(x)≡p\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p(x)\\equiv p$$\\end{document}, i.e., for the p-Laplacian. The fact that our results hold for merely viscosity solutions allows a wide applicability.

Extracting building outlines based on convolutional neural networks using the property of linear connectivity

Abstract. For years, researchers have been developing an automated method that can replace humans by drawing the outlines of individual buildings in a vector format, which plays an important role in GIS creation, environmental monitoring, urban planning, population density estimation, and energy supply. There is no doubt that this is an extremely difficult task, not only because of the labor required to develop such a highly intelligent algorithm, but also because of the challenges posed by imperfect imaging conditions, different building structures, and the complexity of the background. One of the current challenges in extracting building outlines is to accurately recreate the polygonal boundaries of the building while extracting vectorized building masks as output for direct use in various applications. This work provides a comprehensive workflow for building extraction and improves the predicted area of buildings through boundary regularization. First, a convolutional neural network is used to train instance segmentation model, then regularization and vectorization processes are performed. The main difference from existing methods is a new regularization method based on the concepts of linear connectivity and convexity of a set of points. This approach can effectively identify and remove points that do not belong to the detected building but were incorrectly segmented by the algorithm. Based on the results of experiments, the algorithm showed a high level of efficiency, comparable to leading methods for extracting building boundaries as PolyWorld.

Boundary Regularity of Isometries Between Infinitely Flat Complex Domains

Boundary Regularity of Isometries Between Infinitely Flat Complex Domains

Universality for free fermions and the local Weyl law for semiclassical Schrödinger operators

We study local asymptotics for the spectral projector associated to a Schrödinger operator -\hbar^{2}\Delta+V on \mathbb{R}^{n} in the semiclassical limit as \hbar\to0 . We prove local uniform convergence of the rescaled integral kernel of this projector towards a universal model, inside the classically allowed region as well as on its boundary. This implies universality of microscopic fluctuations for the corresponding free fermions (determinantal) point processes, both in the bulk and around regular boundary points. Our results apply to a general class of smooth potentials in arbitrary dimension n\ge 1 . These results are complemented by studying both macroscopic and mesoscopic fluctuations of the point process. We obtain tail bounds for macroscopic linear statistics and, provided n\geq 2 , a central limit theorem for both macroscopic and mesoscopic linear statistics in the bulk.

Oblique Derivative Problem for Parabolic Systems with Quadratic Nonlinearity in Gradient. Boundary Regularity Results

Oblique Derivative Problem for Parabolic Systems with Quadratic Nonlinearity in Gradient. Boundary Regularity Results

Artificial intelligence-based classification of breast nodules: a quantitative morphological analysis of ultrasound images.

Accurate classification of breast nodules into benign and malignant types is critical for the successful treatment of breast cancer. Traditional methods rely on subjective interpretation, which can potentially lead to diagnostic errors. Artificial intelligence (AI)-based methods using the quantitative morphological analysis of ultrasound images have been explored for the automated and reliable classification of breast cancer. This study aimed to investigate the effectiveness of AI-based approaches for improving diagnostic accuracy and patient outcomes. In this study, a quantitative analysis approach was adopted, with a focus on five critical features for evaluation: degree of boundary regularity, clarity of boundaries, echo intensity, and uniformity of echoes. Furthermore, the classification results were assessed using five machine learning methods: logistic regression (LR), support vector machine (SVM), decision tree (DT), naive Bayes, and K-nearest neighbor (KNN). Based on these assessments, a multifeature combined prediction model was established. We evaluated the performance of our classification model by quantifying various features of the ultrasound images and using the area under the receiver operating characteristic (ROC) curve (AUC). The moment of inertia achieved an AUC value of 0.793, while the variance and mean of breast nodule areas achieved AUC values of 0.725 and 0.772, respectively. The convexity and concavity achieved AUC values of 0.988 and 0.987, respectively. Additionally, we conducted a joint analysis of multiple features after normalization, achieving a recall value of 0.98, which surpasses most medical evaluation indexes on the market. To ensure experimental rigor, we conducted cross-validation experiments, which yielded no significant differences among the classifiers under 5-, 8-, and 10-fold cross-validation (P>0.05). The quantitative analysis can accurately differentiate between benign and malignant breast nodules.

Reconstruction of internal heat source in biological tissue using parallel particle swarm optimization

With the rapid development of engineering thermophysics, researches on human biological heat transfer phenomena has gradually shifted from qualitative to quantitative. The heat distribution information of human lesions is of great value for disease analysis, diagnosis, and treatment. It is a typical inverse problem of heat conduction that deriving the distribution of internal heat sources from the temperature distribution on the body surface. Differing from traditional numerical methods for solving heat conduction, this paper transforms such an inverse problem of bio-heat transfer into a direct one, thereby avoiding complex boundary conditions and regularization processes. To noninvasively reconstruct the internal heat source and its corresponding 3D temperature field in biological tissue, the parallel particle swarm optimization (PPSO) is used in the simulation module, where the position P(x, y, z) of point heat source in biological tissue and its corresponding temperature T are set as the optimization variables. Under a certain optimized sample, one can obtain the simulated temperature distributing on the surface of the module, then subtract the simulated temperature from the measured temperature of the same surface which was measured using a thermal infrared imager. If the absolute value of the difference is smaller, it indicates that the current sample is closer to the true location and temperature of the heat source. When the values of optimization variables are determined, the corresponding 3D temperature field is also confirmed. The simulation results show the simulated position and temperature of the heat source are very approximate with those of the real experimental module. The method presented in this paper has enormous potential and promising prospects in clinical research and application, such as tumor hyperthermia, disease thermal diagnosis technology, etc.

Free boundary regularity in the fully nonlinear parabolic thin obstacle problem

Abstract We study the regularity of the free boundary in the fully nonlinear parabolic thin obstacle problem. Under the assumption of time semiconvexity, our main result establishes that the free boundary is a C 1 C^{1} graph in x near any regular free boundary point.

Maximal $L^{1}$-regularity and free boundary problems for the incompressible Navier–Stokes equations in critical spaces

Maximal $L^{1}$-regularity and free boundary problems for the incompressible Navier–Stokes equations in critical spaces

On solvability of the inverse problem for a fourth-order parabolic equation with a complex-valued coefficient

In this paper, the inverse problem for a fourth-order parabolic equation with a variable complex-valued coefficient is studied by the method of separation of variables. The properties of the eigenvalues of the Dirichlet and Neumann boundary value problems for a non-self-conjugate fourth-order ordinary differential equation with a complex-valued coefficient are established. Known results on the Riesz basis property of eigenfunctions of boundary value problems for ordinary differential equations with strongly regular boundary conditions in the space L2(−1,1) are used. On the basis of the Riesz basis property of eigenfunctions, formal solutions of the problems under study are constructed and theorems on the existence and uniqueness of solutions are proved. When proving theorems on the existence and uniqueness of solutions, the Bessel inequality for the Fourier coefficients of expansions of functions from space L2(−1,1) into a Fourier series in the Riesz basis is widely used. The representations of solutions in the form of Fourier series in terms of eigenfunctions of boundary value problems for a fourth-order equation with involution are derived. The convergence of the obtained solutions is discussed.

Some aspects of the Floquet theory for the heat equation in a periodic domain

We treat the linear heat equation in a periodic waveguide Π⊂Rd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Pi \\subset {{\\mathbb {R}}}^d$$\\end{document}, with a regular enough boundary, by using the Floquet transform methods. Applying the Floquet transform F\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\ extsf{F}}}$$\\end{document} to the equation yields a heat equation with mixed boundary conditions on the periodic cell ϖ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varpi $$\\end{document} of Π\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Pi $$\\end{document}, and we analyse the connection between the solutions of the two problems. The considerations involve a description of the spectral projections onto subspaces HS⊂L2(Π)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathcal {H}}}_S \\subset L^2(\\Pi )$$\\end{document} corresponding certain spectral components. We also show that the translated Wannier functions form an orthonormal basis in HS\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathcal {H}}}_S$$\\end{document}.

Remarks on global Hölder regularity for a class of Monge-Ampère type equations

In this paper, we revisit the global Hölder regularity of nonzero convex solutions to the Dirichlet problem for a class of Monge-Ampère type equations. Based on the comparison principle, by choosing delicate auxiliary functions to construct the subsolutions, we extend the boundary regularity obtained in [Li-Li, Sci. China Math. 65 (3) (2022)] to more general equations, in which the parameters α,β,γ in the boundness condition are less restrictive. Also, the non-decreasing condition there can be replaced with a homogeneous type condition.

Holographic Euclidean thermal correlator

In this paper, we compute holographic Euclidean thermal correlators of the stress tensor and U(1) current from the AdS planar black hole. To this end, we set up perturbative boundary value problems for Einstein’s gravity and Maxwell theory in the spirit of Gubser-Klebanov-Polyakov-Witten, with appropriate gauge fixing and regularity boundary conditions at the horizon of the black hole. The linearized Einstein equation and Maxwell equation in the black hole background are related to the Heun equation of degenerate local monodromy. Leveraging the connection relation of local solutions of the Heun equation, we partly solve the boundary value problem and obtain exact two-point thermal correlators for U(1) current and stress tensor in the scalar and shear channels.

The stromal microenvironment endows pancreatic neuroendocrine tumors with spatially specific invasive and metastatic phenotypes

Cancer-associated fibroblasts (CAFs) play an important role in a variety of cancers. However, the role of tumor stroma in nonfunctional pancreatic neuroendocrine tumors (NF-PanNETs) is often neglected. Profiling the heterogeneity of CAFs can reveal the causes of malignant phenotypes in NF-PanNETs. Here, we found that patients with high stromal proportion had poor prognosis, especially for that with infiltrating stroma (stroma and tumor cells that presented an infiltrative growth pattern and no regular boundary). In addition, myofibroblastic CAFs (myCAFs), characterized by FAP+ and α-SMAhigh, were spatially closer to tumor cells and promoted the EMT and tumor growth. Intriguingly, only tumor cells which were spatially closer to myCAFs underwent EMT. We further elucidated that myCAFs stimulate TGF-β expression in nearby tumor cells. Then, TGF-β promoted the EMT in adjacent tumor cells and promoted the expression of myCAFs marker genes in tumor cells, resulting in distant metastasis. Our results indicate that myCAFs cause spatial heterogeneity of EMT, which accounts for liver metastasis of NF-PanNETs. The findings of this study might provide possible targets for the prevention of liver metastasis.

Tracing the disc: The novel qualitative morphometric MRI based disc degeneration classification system.

This study aimed to develop a classification system for lumbar disc degeneration using routine magnetic resonance images (MRIs) that is easily applicable and unaffected by existing classifications' limitations, and to compare its reliability, reproducibility, and discriminative power to the widely used Pfirrmann classification. Five features were graded. This new classification system has eight grades, with at least one of these five features altering each grade. The T2-weighted sagittal images were acquired using a rapid spin-echo sequence with a repetition time of 2680 to 4900 milliseconds, an echo time of 100 to 109 milliseconds, and an echo train length of 17. Slice thick was 4 mm and the display field of view was 32 × 32 cm. The new classification system used five features: signal intensity, disc height, disc boundary regularity, and nucleus annulus separation. Increased signal intensity, decreased height, decreased regularity, and decreased nucleus-annulus separation indicated degeneration. Four raters classified 400 discs from 80 patients using the Pfirrmann and Novel systems. Statistical analyses were conducted to investigate reliability and correlation. The overall ICC and kappa values were found to be higher in the novel classification. (0.988 indicating excellent agreement for ICC and 0.76/0.94 indicating good-very good agreement for kappa). The Kendall tau c value, which shows the correlation between the two classifications and indicates the validity of the new classification, was 0.872, which is very strong. Through the use of cross-tabulations, the discriminatory power of the two newly added classification criteria was determined. This study demonstrates the intra-rater and inter-rater reliability of an easy-to-use, discriminative novel morphometric MRI based classification system for lumbar disc degeneration. The differentiation of grades based on five distinct criteria may generate novel hypotheses regarding treatment selection and response monitoring, as well as new insights into the study of disc degeneration.

A higher order numerical method for singularly perturbed elliptic problems with characteristic boundary layers

A Petrov-Galerkin finite element method is constructed for a singularly perturbed elliptic problem in two space dimensions. The solution contains a regular boundary layer and two characteristic boundary layers. Exponential splines are used as test functions in one coordinate direction and are combined with bilinear trial functions defined on a Shishkin mesh. The resulting numerical method is shown to be a stable parameter-uniform numerical method that achieves a higher order of convergence compared to upwinding on the same mesh.

An Allard-type boundary regularity theorem for 2d\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$2d$\\end{document} minimizing currents at smooth curves with arbitrary multiplicity

We consider integral area-minimizing 2-dimensional currents T\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$T$\\end{document} in U⊂R2+n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$U\\subset \\mathbf {R}^{2+n}$\\end{document} with ∂T=Q〚Γ〛\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\partial T = Q\\left [\\!\\![{\\Gamma }\\right ]\\!\\!]$\\end{document}, where Q∈N∖{0}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$Q\\in \\mathbf {N} \\setminus \\{0\\}$\\end{document} and Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\Gamma $\\end{document} is sufficiently smooth. We prove that, if q∈Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$q\\in \\Gamma $\\end{document} is a point where the density of T\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$T$\\end{document} is strictly below Q+12\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\frac{Q+1}{2}$\\end{document}, then the current is regular at q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$q$\\end{document}. The regularity is understood in the following sense: there is a neighborhood of q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$q$\\end{document} in which T\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$T$\\end{document} consists of a finite number of regular minimal submanifolds meeting transversally at Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\Gamma $\\end{document} (and counted with the appropriate integer multiplicity). In view of well-known examples, our result is optimal, and it is the first nontrivial generalization of a classical theorem of Allard for Q=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$Q=1$\\end{document}. As a corollary, if Ω⊂R2+n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\Omega \\subset \\mathbf {R}^{2+n}$\\end{document} is a bounded uniformly convex set and Γ⊂∂Ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\Gamma \\subset \\partial \\Omega $\\end{document} a smooth 1-dimensional closed submanifold, then any area-minimizing current T\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$T$\\end{document} with ∂T=Q〚Γ〛\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\partial T = Q \\left [\\!\\![{\\Gamma }\\right ]\\!\\!]$\\end{document} is regular in a neighborhood of Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\Gamma $\\end{document}.

Trace Operator on von Koch’s Snowflake

We study properties of the boundary trace operator on the Sobolev space W11(Ω)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$W^1_1(\\Omega )$$\\end{document}. Using the density result by Koskela and Zhang (Arch. Ration. Mech. Anal. 222(1), 1-14 2016), we define a surjective operator Tr:W11(ΩK)→X(ΩK)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Tr: W^1_1(\\Omega _K)\\rightarrow X(\\Omega _K)$$\\end{document}, where ΩK\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega _K$$\\end{document} is von Koch’s snowflake and X(ΩK)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$X(\\Omega _K)$$\\end{document} is a trace space with the quotient norm. Since ΩK\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega _K$$\\end{document} 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(ΩK)→W11(ΩK)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S: X(\\Omega _K) \\rightarrow W^1_1(\\Omega _K)$$\\end{document} such that Tr∘S=IdX(ΩK)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Tr \\circ S= Id_{X(\\Omega _K)}$$\\end{document}. 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 ℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _1$$\\end{document}. 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 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} with regular boundary, which explains Banach space geometry cause for this phenomenon.