Search results for "Boundary condition"
showing 10 items of 235 documents
Dirichlet Boundary Value Problem for the Second Order Asymptotically Linear System
2016
We consider the second order system x′′=f(x) with the Dirichlet boundary conditions x(0)=0=x(1), where the vector field f∈C1(Rn,Rn) is asymptotically linear and f(0)=0. We provide the existence and multiplicity results using the vector field rotation theory.
On vibrating thin membranes with mass concentrated near the boundary: an asymptotic analysis
2018
We consider the spectral problem \begin{equation*} \left\{\begin{array}{ll} -\Delta u_{\varepsilon}=\lambda(\varepsilon)\rho_{\varepsilon}u_{\varepsilon} & {\rm in}\ \Omega\\ \frac{\partial u_{\varepsilon}}{\partial\nu}=0 & {\rm on}\ \partial\Omega \end{array}\right. \end{equation*} in a smooth bounded domain $\Omega$ of $\mathbb R^2$. The factor $\rho_{\varepsilon}$ which appears in the first equation plays the role of a mass density and it is equal to a constant of order $\varepsilon^{-1}$ in an $\varepsilon$-neighborhood of the boundary and to a constant of order $\varepsilon$ in the rest of $\Omega$. We study the asymptotic behavior of the eigenvalues $\lambda(\varepsilon)$ and the eige…
Multiplicity of Solutions for Second Order Two-Point Boundary Value Problems with Asymptotically Asymmetric Nonlinearities at Resonance
2007
Abstract Estimations of the number of solutions are given for various resonant cases of the boundary value problem 𝑥″ + 𝑔(𝑡, 𝑥) = 𝑓(𝑡, 𝑥, 𝑥′), 𝑥(𝑎) cos α – 𝑥′(𝑎) sin α = 0, 𝑥(𝑏) cos β – 𝑥′(𝑏) sin β = 0, where 𝑔(𝑡, 𝑥) is an asymptotically linear nonlinearity, and 𝑓 is a sublinear one. We assume that there exists at least one solution to the BVP.
Finite Size Effects in Thin Film Simulations
2003
Phase transitions in thin films are discussed, with an emphasis on Ising-type systems (liquid-gas transition in slit-like pores, unmixing transition in thin films, orderdisorder transitions on thin magnetic films, etc.) The typical simulation geometry then is a L xL x D system, where at the low confining L x L surfaces appropriate boundary “fields” are applied, while in the lateral directions periodic boundary conditions are used. In the z-direction normal to the film, the order parameter always is inhomogeneous, due to the boundary “fields” at the confining surfaces. When one varies the temperature T from the region of the bulk disordered phase to a temperature below the critical temperatu…
Stress fields by the symmetric Galerkin boundary element method
2004
The paper examines the stress state of a body with the discretized boundary embedded in the infinite domain subjected to layered or double-layered actions, such as forces and displacement discontinuities on the boundary, and to internal actions, such as body forces and thermic variations, in the ambit of the symmetric Galerkin boundary element method (SGBEM). The stress distributions due to internal actions (body forces and thermic variations) were computed by transforming the volume integrals into boundary integrals. The aim of the paper is to show the tension state in Ω∞ as a response to all the actions acting in Ω when this analysis concerns the crossing of the discretized boundary, thu…
Stress gradient versus strain gradient constitutive models within elasticity
2014
Abstract A stress gradient elasticity theory is developed which is based on the Eringen method to address nonlocal elasticity by means of differential equations. By suitable thermodynamics arguments (involving the free enthalpy instead of the free internal energy), the restrictions on the related constitutive equations are determined, which include the well-known Eringen stress gradient constitutive equations, as well as the associated (so far uncertain) boundary conditions. The proposed theory exhibits complementary characters with respect to the analogous strain gradient elasticity theory. The associated boundary-value problem is shown to admit a unique solution characterized by a Helling…
Two -methods to generate Bézier surfaces from the boundary
2009
Two methods to generate tensor-product Bezier surface patches from their boundary curves and with tangent conditions along them are presented. The first one is based on the tetraharmonic equation: we show the existence and uniqueness of the solution of @D^4x->=0 with prescribed boundary and adjacent to the boundary control points of a nxn Bezier surface. The second one is based on the nonhomogeneous biharmonic equation @D^2x->=p, where p could be understood as a vectorial load adapted to the C^1-boundary conditions.
Fractional p-Laplacian evolution equations
2016
Abstract In this paper we study the fractional p-Laplacian evolution equation given by u t ( t , x ) = ∫ A 1 | x − y | N + s p | u ( t , y ) − u ( t , x ) | p − 2 ( u ( t , y ) − u ( t , x ) ) d y for x ∈ Ω , t > 0 , 0 s 1 , p ≥ 1 . In a bounded domain Ω we deal with the Dirichlet problem by taking A = R N and u = 0 in R N ∖ Ω , and the Neumann problem by taking A = Ω . We include here the limit case p = 1 that has the extra difficulty of giving a meaning to u ( y ) − u ( x ) | u ( y ) − u ( x ) | when u ( y ) = u ( x ) . We also consider the Cauchy problem in the whole R N by taking A = Ω = R N . We find existence and uniqueness of strong solutions for each of the above mentioned problem…
A logarithmic fourth-order parabolic equation and related logarithmic Sobolev inequalities
2006
A logarithmic fourth-order parabolic equation in one space dimension with periodic boundary conditions is studied. This equation arises in the context of fluctuations of a stationary nonequilibrium interface and in the modeling of quantum semiconductor devices. The existence of global-in-time non-negative weak solutions and some regularity results are shown. Furthermore, we prove that the solution converges exponentially fast to its mean value in the ``entropy norm'' and in the Fisher information, using a new optimal logarithmic Sobolev inequality for higher derivatives. In particular, the rate is independent of the solution and the constant depends only on the initial value of the entropy.
Solution of a cauchy problem for an infinite chain of linear differential equations
2005
Defining the recurrence relations for orthogonal polynomials we have found an exact solution of a Cauchy problem for an infinite chain of linear differential equations with constant coefficients. These solutions have been found both for homogeneous and an inhomogeneous systems.