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1

Singular elliptic problems with Dirichlet or mixed Dirichlet-Neumann non-homogeneous boundary conditions

Godoy Tomas

Opuscula Mathematica

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2023

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Vol. 43, no. 1

19--46

EN

Let Ω be a C2 bounded domain in Rn such that ∂Ω = Γ1 ∪ Γ2, where Γ1 and Γ2 are disjoint closed subsets of ∂Ω, and consider the problem −Δu = g(·, u) in Ω, u = τ on Γ1, ∂u ∂ν = η on Γ2, where 0 ≤ τ ∈ W 1 2 ,2(Γ1), η ∈ (H1 0, Γ1(Ω))′, and g : Ω×(0,∞) → R is a nonnegative Carathéodory function. Under suitable assumptions on g and η we prove the existence and uniqueness of a positive weak solution of this problem. Our assumptions allow g to be singular at s = 0 and also at x ∈ S for some suitable subsets S ⊂ Ω. The Dirichlet problem −Δu = g(·, u) in Ω, u = σ on ∂Ω is also studied in the case when 0 ≤ σ ∈ W 1 2 ,2(Ω).

2

Neumann boundary optimal control problems governed by parabolic variational equalities

Bollo Carolina M., Gariboldi Claudia M., Tarzia Domingo A.

Control and Cybernetics

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2021

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Vol. 50, No. 2

227--252

EN

We consider a heat conduction problem S with mixed boundary conditions in an n-dimensional domain with regular boundary and a family of problems Sα with also mixed boundary conditions in , where α > 0 is the heat transfer coefficient on the portion of the boundary Г1. In relation to these state systems, we formulate Neumann boundary optimal control problems on the heat flux q which is definite on the complementary portion Г2 of the boundary of Ω. We obtain existence and uniqueness of the optimal controls, the first order optimality conditions in terms of the adjoint state and the convergence of the optimal controls, the system state and the adjoint state when the heat transfer coefficient α goes to infinity. Furthermore, we formulate particular boundary optimal control problems on a real parameter λ, in relation to the parabolic problems S and Sαα

3

Ritz method for large deflection of orthotropic thin plates with mixed boundary conditions

Al-Shugaa Madyan A., Al-Gahtani Husain J., Musa Abubakr E.S.

Journal of Applied Mathematics and Computational Mechanics

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2020

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Vol. 19, nr 2

5--16

EN

In this paper, the Ritz method is developed for the analysis of thin rectangular orthotropic plates undergoing large deflection. The trial functions approximating the plate lateral and in-plane displacements are represented by simple polynomials. The nonlinear algebraic equations resulting from the application of the concept of minimum potential energy of the orthotropic plate are cast in a matrix form. The developed matrix form equations are then implemented in a Mathematica code that allows for the automation of the solution for an arbitrary number of the trial polynomials. The developed code is tested through several numerical examples involving rectangular plates with different aspect ratios and boundary conditions. The results of all examples demonstrate the efficiency and accuracy of the proposed method.

4

Existence and uniqueness of a problem in thermo-elasto-plasticity with phase transitions in TRIP steels under mixed boundary conditions

Boettcher S.

Journal of Applied Analysis

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2018

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Vol. 24, nr 1

87--98

EN

In this paper a complex model describing thermo-elasto-plasticity, phase transitions (PT) and transformation-induced plasticity (TRIP) is studied. The main objective is the analysis of the corresponding initial and boundary value problem (IBVP) considering linearized thermo-elastic dissipation and a viscosity-like regularization.

5

Influence of mixed boundary conditions and heterogeneity on the vibration behavior of orthotropic truncated conical shells

Sofiyev A. H., Huseynov S. E., Kuruoglu N.

Archives of Mechanics

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2015

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Vol. 67, nr 4

331--348

EN

This paper presents the vibration behavior analysis of heterogeneous orthotropic conical shells with mixed boundary conditions. Basic equations of heterogeneous orthotropic truncated conical shells are derived using Donnell–Mushtari shell theory. Employing the separation of variables and Galerkin’s method, the expressions for frequency of heterogeneous orthotropic conical shells with two mixed boundary conditions are obtained. The results are validated through numerical comparisons with available results in the literature. The influences of truncated shell characteristics, heterogeneity, material orthotropy and mixed boundary conditions on dimensionless frequency parameters are investigated.

6

Crack nucleation in circular disk under mixed boundary conditions

Mirsalimov V. M., Kalantarly N. M.

Archives of Mechanics

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2015

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Vol. 67, nr 2

115--136

EN

A model of crack nucleation in a circular disk, based on consideration of cracking process zone is suggested. It is assumed that the cracking process zone is a finitelength layer containing a material with partially disturbed bonds between separate structural elements. Existence of bonds between the pre-fracture zone faces (the area of weakened interparticle bonds of the material) is simulated by application of cohesive forces caused by the existence of bonds to pre-fracture area surfaces. Analysis of limit equilibrium of the pre-fracture zone in a circular disk with mixed conditions on the boundary are fulfilled on the basis of ultimate stretching of material’s bonds and includes: 1) setting up the dependence of cohesive forces on opening of pre-fracture area faces, 2) estimation of stress state near the pre-fracture zone with regard to external loads and cohesive forces, 3) determination of dependence of critical external loads on geometrical parameters of the disk, under which the crack appears.