scholarly journals Properties of the free boundary near the fixed boundary of the double obstacle problems

2021 ◽  
Author(s):  
Jinwan Park
Keyword(s):  
Free Boundary ◽  
Obstacle Problem ◽  
Nonlinear Operator ◽  
Main Idea ◽  
Global Solutions ◽  
Obstacle Problems ◽  
Local Regularity ◽  
Fixed Boundary ◽  
New Type

In this paper, we study the tangential touch and [Formula: see text] regularity of the free boundary near the fixed boundary of the double obstacle problem for Laplacian and fully nonlinear operator. The main idea to have the properties is regarding the upper obstacle as a solution of the single obstacle problem. Then, in the classification of global solutions of the double problem, it is enough to consider only two cases for the upper obstacle, [Formula: see text] The second one is a new type of upper obstacle, which does not exist in the study of local regularity of the free boundary of the double problem. Thus, in this paper, a new type of difficulties that come from the second type upper obstacle is mainly studied.

scholarly journals The local structure of the free boundary in the fractional obstacle problem

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Matteo Focardi ◽  
Emanuele Spadaro
Keyword(s):  
Hausdorff Dimension ◽  
Free Boundary ◽  
Hausdorff Measure ◽  
Local Structure ◽  
Obstacle Problem ◽  
List Type ◽  
Minkowski Content ◽  
Local Finiteness ◽  
Lower Dimensional

AbstractBuilding upon the recent results in [M. Focardi and E. Spadaro, On the measure and the structure of the free boundary of the lower-dimensional obstacle problem, Arch. Ration. Mech. Anal. 230 2018, 1, 125–184] we provide a thorough description of the free boundary for the solutions to the fractional obstacle problem in {\mathbb{R}^{n+1}} with obstacle function φ (suitably smooth and decaying fast at infinity) up to sets of null {{\mathcal{H}}^{n-1}} measure. In particular, if φ is analytic, the problem reduces to the zero obstacle case dealt with in [M. Focardi and E. Spadaro, On the measure and the structure of the free boundary of the lower-dimensional obstacle problem, Arch. Ration. Mech. Anal. 230 2018, 1, 125–184] and therefore we retrieve the same results:(i)local finiteness of the {(n-1)}-dimensional Minkowski content of the free boundary (and thus of its Hausdorff measure),(ii){{\mathcal{H}}^{n-1}}-rectifiability of the free boundary,(iii)classification of the frequencies and of the blowups up to a set of Hausdorff dimension at most {(n-2)} in the free boundary.Instead, if {\varphi\in C^{k+1}(\mathbb{R}^{n})}, {k\geq 2}, similar results hold only for distinguished subsets of points in the free boundary where the order of contact of the solution with the obstacle function φ is less than {k+1}.

scholarly journals Boundary Regularity for p-Harmonic Functions and Solutions of Obstacle Problems on Unbounded Sets in Metric Spaces

2019 ◽  
Vol 7 (1) ◽  
pp. 179-196
Author(s):  
Anders Björn ◽  
Daniel Hansevi
Keyword(s):  
Harmonic Functions ◽  
Obstacle Problem ◽  
Metric Spaces ◽  
Local Property ◽  
Boundary Regularity ◽  
Obstacle Problems ◽  
Regular Boundary ◽  
Complete Metric Spaces ◽  
Regularity Results

Abstract The theory of boundary regularity for p-harmonic functions is extended to unbounded open sets in complete metric spaces with a doubling measure supporting a p-Poincaré inequality, 1 < p < ∞. The barrier classification of regular boundary points is established, and it is shown that regularity is a local property of the boundary. We also obtain boundary regularity results for solutions of the obstacle problem on open sets, and characterize regularity further in several other ways.

Petrology and classification of A‐9003, A 09535, and Y‐82094: A new type of carbonaceous chondrite

2021 ◽  
Author(s):  
M. Kimura ◽  
R. C. Greenwood ◽  
M. Komatsu ◽  
N. Imae ◽  
A. Yamaguchi ◽  
...  
Keyword(s):  
Carbonaceous Chondrite ◽  
New Type

scholarly journals Higher differentiability results for solutions to a class of non-autonomous obstacle problems with sub-quadratic growth conditions

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Andrea Gentile
Keyword(s):  
Besov Space ◽  
Lipschitz Space ◽  
Growth Conditions ◽  
Obstacle Problems ◽  
Quadratic Growth ◽  
Regularity Assumption ◽  
Lipschitz Regularity ◽  
Fixed Boundary ◽  
Higher Differentiability ◽  
Admissible Functions

Abstract We establish some higher differentiability results of integer and fractional order for solutions to non-autonomous obstacle problems of the form min ⁡ { ∫ Ω f ⁢ ( x , D ⁢ v ⁢ ( x ) ) : v ∈ K ψ ⁢ ( Ω ) } , \min\biggl{\{}\int_{\Omega}f(x,Dv(x)):v\in\mathcal{K}_{\psi}(\Omega)\biggr{\}}, where the function 𝑓 satisfies 𝑝-growth conditions with respect to the gradient variable, for 1 < p < 2 1<p<2 , and K ψ ⁢ ( Ω ) \mathcal{K}_{\psi}(\Omega) is the class of admissible functions v ∈ u 0 + W 0 1 , p ⁢ ( Ω ) v\in u_{0}+W^{1,p}_{0}(\Omega) such that v ≥ ψ v\geq\psi a.e. in Ω, where u 0 ∈ W 1 , p ⁢ ( Ω ) u_{0}\in W^{1,p}(\Omega) is a fixed boundary datum. Here we show that a Sobolev or Besov–Lipschitz regularity assumption on the gradient of the obstacle 𝜓 transfers to the gradient of the solution, provided the partial map x ↦ D ξ ⁢ f ⁢ ( x , ξ ) x\mapsto D_{\xi}f(x,\xi) belongs to a suitable Sobolev or Besov space. The novelty here is that we deal with sub-quadratic growth conditions with respect to the gradient variable, i.e. f ⁢ ( x , ξ ) ≈ a ⁢ ( x ) ⁢ | ξ | p f(x,\xi)\approx a(x)\lvert\xi\rvert^{p} with 1 < p < 2 1<p<2 , and where the map 𝑎 belongs to a Sobolev or Besov–Lipschitz space.

scholarly journals MultiKOC: Multi-One-Class Classifier Based K-Means Clustering

Algorithms ◽  
2021 ◽  
Vol 14 (5) ◽  
pp. 134
Author(s):  
Loai Abdallah ◽  
Murad Badarna ◽  
Waleed Khalifa ◽  
Malik Yousef
Keyword(s):  
Clustering Algorithm ◽  
Main Idea ◽  
Molecular Classification ◽  
Positive Sample ◽  
Classification Problems ◽  
Multiple Cancer ◽  
Multiple Tumor ◽  
One Class Classifier ◽  
The Given

In the computational biology community there are many biological cases that are considered as multi-one-class classification problems. Examples include the classification of multiple tumor types, protein fold recognition and the molecular classification of multiple cancer types. In all of these cases the real world appropriately characterized negative cases or outliers are impractical to achieve and the positive cases might consist of different clusters, which in turn might lead to accuracy degradation. In this paper we present a novel algorithm named MultiKOC multi-one-class classifiers based K-means to deal with this problem. The main idea is to execute a clustering algorithm over the positive samples to capture the hidden subdata of the given positive data, and then building up a one-class classifier for every cluster member’s examples separately: in other word, train the OC classifier on each piece of subdata. For a given new sample, the generated classifiers are applied. If it is rejected by all of those classifiers, the given sample is considered as a negative sample, otherwise it is a positive sample. The results of MultiKOC are compared with the traditional one-class, multi-one-class, ensemble one-classes and two-class methods, yielding a significant improvement over the one-class and like the two-class performance.

scholarly journals Special Finite Elements with Adaptive Strain Field on the Example of One-Dimensional Elements

2021 ◽  
Vol 11 (2) ◽  
pp. 609
Author(s):  
Tadeusz Chyży ◽  
Monika Mackiewicz
Keyword(s):  
Finite Elements ◽  
Strain Field ◽  
Main Idea ◽  
Deformation Field ◽  
Geometrical Parameters ◽  
Single Element ◽  
One Dimensional ◽  
New Type ◽  
Self Adaptation ◽  
Adaptation Method

The conception of special finite elements called multi-area elements for the analysis of structures with different stiffness areas has been presented in the paper. A new type of finite element has been determined in order to perform analyses and calculations of heterogeneous, multi-coherent, and layered structures using fewer finite elements and it provides proper accuracy of the results. The main advantage of the presented special multi-area elements is the possibility that areas of the structure with different stiffness and geometrical parameters can be described by single element integrated in subdivisions (sub-areas). The formulation of such elements has been presented with the example of one-dimensional elements. The main idea of developed elements is the assumption that the deformation field inside the element is dependent on its geometry and stiffness distribution. The deformation field can be changed and adjusted during the calculation process that is why such elements can be treated as self-adaptive. The application of the self-adaptation method on strain field should simplify the analysis of complex non-linear problems and increase their accuracy. In order to confirm the correctness of the established assumptions, comparative analyses have been carried out and potential areas of application have been indicated.

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