scholarly journals On the boundary regularity of phase-fields for Willmore's energy

2018 ◽  
Vol 149 (04) ◽  
pp. 1017-1035
Author(s):  
Patrick W. Dondl ◽  
Stephan Wojtowytsch
Keyword(s):  
Boundary Conditions ◽  
Partial Regularity ◽  
Boundary Regularity ◽  
Radon Measures ◽  
Practical Applications ◽  
Phase Fields ◽  
Regularity Results ◽  
Willmore Energy ◽  
Uniformly Bounded

AbstractWe demonstrate that Radon measures which arise as the limit of the Modica-Mortola measures associated with phase-fields with uniformly bounded diffuse area and Willmore energy may be singular at the boundary of a domain and discuss implications for practical applications. We furthermore give partial regularity results for the phase-fields uε at the boundary in terms of boundary conditions and counterexamples without boundary conditions.

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.

scholarly journals A neutral core of degressively proportional allocations under lexicographic preferences of agents

2021 ◽  
Author(s):  
Katarzyna Cegiełka ◽  
Piotr Dniestrzański ◽  
Janusz Łyko ◽  
Arkadiusz Maciuk ◽  
Maciej Szczeciński
Keyword(s):  
European Union ◽  
Numerical Analysis ◽  
Theoretical Analysis ◽  
Boundary Conditions ◽  
Case Studies ◽  
The European Union ◽  
Practical Applications ◽  
Lexicographic Preferences ◽  
Degree Of Acceptance ◽  
Feasible Solutions

AbstractOne of the main problems of practical applications of degressively proportional allocations of goods and burdens is lack of uniqueness of this principle. Even under given boundary conditions of allocation, i.e. determined minimal and maximal amounts of a good that can be assigned in a given allocation, there are usually many feasible solutions. The lack of formal rules of allocation is the reason why the allocation is typically a result of negotiations among its agents. A number of allocations favor some of agents or their groups, therefore other agents cannot accept them. The aim of this paper is to indicate a way of reducing the set of all feasible solutions exclusively to those that are neutral to all agents. As a result of the term of lexicographic preference of allocation agents defined on the basis of the relation theory followed by a numerical analysis of sets of all feasible solutions, it is possible to determine a core of this set in the form of a subset of all feasible solutions that are acceptable by all agents. In addition, this subset can be further divided into smaller subsets with regard to the degree of acceptance of their elements. Theoretical analysis is complemented by case studies, one of which is application of this idea to the allocation of seats in the European Parliament among the member states of the European Union.

Boundary Conditions and Sensitivities of Mass-Elastic Systems

1988 ◽  
Vol 110 (4) ◽  
pp. 464-467
Author(s):  
D. Morrison
Keyword(s):  
Boundary Conditions ◽  
External Disturbance ◽  
Practical Interest ◽  
Short Paper ◽  
Practical Applications ◽  
General Terms ◽  
Elastic Systems ◽  
Seismic Applications

It has been noted in numerous practical applications that the sensitivities of mass elastic systems to external disturbance are closely bound up with the nonzero values of the normalized boundary and field actions, both forces and displacements. This leads in many cases to rapid analyses without, for example, the need for a separate calculation of “participation factors” in seismic applications. This short paper sets down some relationships in general terms and indicates examples of practical interest.

scholarly journals A simple method for implementing Monte Carlo tests

2019 ◽  
Vol 35 (3) ◽  
pp. 1373-1392 ◽  
Author(s):  
Dong Ding ◽  
Axel Gandy ◽  
Georg Hahn
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Simulations ◽  
Statistical Test ◽  
Uniform Bound ◽  
Data Set ◽  
Simple Method ◽  
Practical Applications ◽  
Tuning Parameters ◽  
Sequence Method ◽  
Uniformly Bounded

Abstract We consider a statistical test whose p value can only be approximated using Monte Carlo simulations. We are interested in deciding whether the p value for an observed data set lies above or below a given threshold such as 5%. We want to ensure that the resampling risk, the probability of the (Monte Carlo) decision being different from the true decision, is uniformly bounded. This article introduces a simple open-ended method with this property, the confidence sequence method (CSM). We compare our approach to another algorithm, SIMCTEST, which also guarantees an (asymptotic) uniform bound on the resampling risk, as well as to other Monte Carlo procedures without a uniform bound. CSM is free of tuning parameters and conservative. It has the same theoretical guarantee as SIMCTEST and, in many settings, similar stopping boundaries. As it is much simpler than other methods, CSM is a useful method for practical applications.

Evaluation of the influence of boundary confinement on the behaviour of unsaturated swelling clay soils

2009 ◽  
Vol 46 (3) ◽  
pp. 339-356 ◽  
Author(s):  
Greg Siemens ◽  
James A. Blatz
Keyword(s):  
Boundary Conditions ◽  
Volume Expansion ◽  
Experimental Testing ◽  
Mean Stress ◽  
Equilibrium Limit ◽  
Practical Applications ◽  
Swelling Soils ◽  
Liquid Infiltration ◽  
Volume State ◽  
The Impact

Swelling soils are found in many regions throughout the world. Damage caused to infrastructure by these types of soils is measured annually in billions of dollars. These excessive damages are, in part, due to the lack of proper design, resulting from a need for better tools for practitioners to assess the impact of swelling soils in typical design applications. This paper presents an experimental testing program with interpretations to provide a framework for predicting the behaviour of swelling soils under general stress and volume state conditions for practical applications. The experimental testing adopted a new automated triaxial apparatus that controls boundary stress and strain while applying liquid infiltration conditions at the perimeter or center of triaxial specimens. Results demonstrate the influence of a range of boundary conditions on the behaviour of swelling soil during liquid infiltration. The range of boundary conditions examined in the experimental testing include constant mean stress (CMS), where the mean stress applied during the swelling stage is constant; constant volume (CV), where the volume is held constant during the liquid infiltration; as well as a flexible spring-type boundary condition (CS) that applies increases in stress as a specified function of the volume increase. These boundary conditions represent the broad spectrum of experiences in the field. The experimental results show the dominance of boundary conditions on the development of swell pressure and volume expansion to give evidence for a new swell equilibrium limit (SEL) relationship. The SEL shows promise in providing a framework for swelling soils to predict the final soil state under wetting conditions for the range of boundary conditions examined. Application of the SEL relationship in practice is presented as a concept for examining swelling induced pressures and volume expansion in applications of liquid infiltration of swelling soils.

scholarly journals Regularity results for singular elliptic problems

2006 ◽  
Vol 4 (3) ◽  
pp. 243-259 ◽  
Author(s):  
Loredana Caso
Keyword(s):  
Boundary Conditions ◽  
Weight Function ◽  
Elliptic Equations ◽  
Global Regularity ◽  
Elliptic Problems ◽  
The Other ◽  
Weighted Spaces ◽  
Regularity Results

Some local and global regularity results for solutions of linear elliptic equations in weighted spaces are proved. Here the leading coefficients are VMO functions, while the hypotheses on the other coefficients and the boundary conditions involve a suitable weight function.

scholarly journals Factor spaces and implications on Kirchhoff equations with clamped boundary conditions

2001 ◽  
Vol 6 (8) ◽  
pp. 441-488 ◽  
Author(s):  
Irena Lasiecka ◽  
Roberto Triggiani
Keyword(s):  
Boundary Conditions ◽  
Boundary Control ◽  
Exact Controllability ◽  
Elastic Displacement ◽  
Kirchhoff Equations ◽  
Quotient Spaces ◽  
Mixed Problems ◽  
Duality Argument ◽  
Regularity Results ◽  
New Phenomena

We consider mixed problems for the Kirchhoff elastic and thermoelastic systems, subject to boundary control in the clamped boundary conditions BC (clamped control). Ifwdenotes the elastic displacement andθthe temperature, we establish sharp regularity of{w,wt,wtt}in the elastic case, and of{w,wt,wtt,θ}in the thermoelastic case. Our results complement those by Lagnese and Lions (1988), where sharp (optimal) trace regularity results are obtained for the corresponding boundary homogeneous cases. The passage from the boundary homogeneous cases to the corresponding mixed problems involves a duality argument. However, in the present case of clamped BC, and only in this case, the duality argument in question is both delicate and technical. In this respect, the clamped BC are “exceptional” within the set of canonical BC (hinged, clamped, free BC). Indeed, it produces new phenomena which are accounted for by introducing new, untraditional factor (quotient) spaces. These are critical in describing both interior regularity and exact controllability of mixed elastic and thermoelastic Kirchhoff problems with clamped controls.

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