155 research outputs found
Results on Nonlocal Boundary Value Problems
No full textIn this article, we provide a variational theory for nonlocal problems where nonlocality arises due to the interaction in a given horizon. With this theory, we prove well-posedness results for the weak formulation of nonlocal boundary value problems with Dirichlet, Neumann, and mixed boundary conditions for a class of kernel functions. The motivating application for nonlocal boundary value problems is the scalar stationary peridynamics equation of motion. The well-posedness results support practical kernel functions used in the peridynamics setting. We also prove a spectral equivalence estimate which leads to a mesh size independent upper bound for the condition number of an underlying discretized operator. This is a fundamental conditioning result that would guide preconditioner construction for nonlocal problems. The estimate is a consequence of a nonlocal Poincare-type inequality that reveals a horizon size quantification. We provide an example that establishes the sharpness of the upper bound in the spectral equivalence.B. Aksoylu was supported in part by the NSF DMS-1016190 grant and T. Mengesha was supported by the NSF DMS-0406374 grant.NSFNational Science Foundation (NSF) [DMS-1016190, DMS-0406374
Solvability of nonlocal systems related to peridynamics
No full textKaßmann M, Mengesha T, Scott J. Solvability of nonlocal systems related to peridynamics. Communications on Pure and Applied Analysis . 2019;18(3):1303-1332.In this work, we study the Dirichlet problem associated with a strongly coupled system of nonlocal equations. The system of equations comes from a linearization of a model of peridynamics, a nonlocal model of elasticity. It is a nonlocal analogue of the Navier-Lame system of classical elasticity. The leading operator is an integro-differential operator characterized by a distinctive matrix kernel which is used to couple differences of components of a vector field. The paper's main contributions are proving well-posedness of the system of equations and demonstrating optimal local Sobolev regularity of solutions. We apply Hilbert space techniques for well-posedness. The result holds for systems associated with kernels that give rise to non-symmetric bilinear forms. The regularity result holds for systems with symmetric kernels that may be supported only on a cone. For some specific kernels associated energy spaces are shown to coincide with standard fractional Sobolev spaces
Representation formulas for
No full textWe examine the composition of the L∞ norm with weakly
convergent sequences of gradient fields associated with the homogenization of second order
divergence form partial differential equations with measurable coefficients. Here the
sequences of coefficients are chosen to model heterogeneous media and are piecewise
constant and highly oscillatory. We identify local representation formulas that in the
fine phase limit provide upper bounds on the limit superior of the
L∞ norms of gradient fields. The local representation
formulas are expressed in terms of the weak limit of the gradient fields and local
corrector problems. The upper bounds may diverge according to the presence of rough
interfaces. We also consider the fine phase limits for layered microstructures and for
sufficiently smooth periodic microstructures. For these cases we are able to provide
explicit local formulas for the limit of the L∞ norms of the
associated sequence of gradient fields. Local representation formulas for lower bounds are
obtained for fields corresponding to continuously graded periodic microstructures as well
as for general sequences of oscillatory coefficients. The representation formulas are
applied to problems of optimal material design
Global Estimates for Quasilinear Elliptic Equations on Reifenberg Flat Domains
No full textNew global regularity estimates are obtained for solutions to a class of quasilinear elliptic boundary value problems. The coefficients are assumed to have small BMO seminorms, and the boundary of the domain is sufficiently flat in the sense of Reifenberg. The main regularity estimates obtained are in weighted Lorentz spaces. Other regularity results in Lorentz-Morrey, Morrey, and Hölder spaces are shown to follow from the main estimates. © 2011 Springer-Verlag
Weighted and regularity estimates for nonlinear equations on Reifenberg flat domains
No full textAbstractGlobal weighted Lp estimates are obtained for the gradient of solutions to nonlinear elliptic Dirichlet boundary value problems over a bounded nonsmooth domain. Morrey and Hölder regularity of solutions are also established, as a consequence. These results generalize various existing estimates for nonlinear equations. The nonlinearities are of at most linear growth and assumed to have a uniform small mean oscillation. The boundary of the domain, on the other hand, may exhibit roughness but assumed to be sufficiently flat in the sense of Reifenberg. Our approach uses maximal function estimates and Vitali covering lemma, and also known regularity results of solutions to nonlinear homogeneous equations
Quasilinear Riccati type equations with distributional data in Morrey space framework
No full textIn the framework of Morrey or Lorentz-Morrey spaces, we characterize the existence of solutions to the quasilinear Riccati type equation. with a distributional datum σ. Here div A(x, ∇u) is a quasilinear elliptic operator modelled after the p-Laplacian, p \u3e. 1, but with a very general nonlinear structure, and Ω is a sufficiently flat domain in the sense of Reifenberg. The existence results are obtained in the natural or super-natural range of the gradient growth, i.e., q≥. p.Motivated by the analysis of quasilinear Riccati type equation, a substantial part of the paper is also devoted to the Calderón-Zygmund type gradient regularity for the boundary value problem. We obtain regularity estimates in some weighted and unweighted function spaces as well as natural Lorentz-Morrey spaces associated to the Riccati type equation above
Representation formulas for norms of weakly convergent sequences of gradient fields in homogenization
No full text
Characterization of function spaces of vector fields and an application in nonlinear peridynamics
No full textAssessment on bankruptcy analysis of commercial bank of Ethiopia
No full textBankruptcy is a state of insolvency where in the company or the person is not able to repay the
creditors the debt amount. Bankruptcy prediction is of importance to the various stakeholders of
the company as well as the society on the whole. The purpose of my research is study the
analysis bankruptcy and sustainability position of commercial banks in the Ameya branch. Many
Banking sector, which will been declared sick. The researcher used judgmental sampling
techniques. The research shall make analysis by distributing structured questioners developed by
researcher for individual respondents. The researcher concluded that high level of debt position,
low level of liquidity ratio, low level of leverage ratio, low earnings before interest and tax are
the main cause of bankruptcy in commercial Bank of Ethiopia Amaya branch and finally the
researcher recommended that minimizing the firms leverage through financial restructuring,
improving the efficiency of the firm asset through reduction of expenditure are assumed by the
researcher
- …
