We prove the convergence of an adaptive linear finite element method for computing eigenvalues and eigenfunctions of second order symmetric elliptic partial differential operators. The weak form is assumed to yield a bilinear form which is bounded and coercive in H. Each step of the adaptive procedure refines elements in which a standard a posteriori error estimator is large and also refines el...

this paper is concerned with the study of the existence of positive solutions for a navier boundaryvalue problem involving the p-biharmonic operator; the right hand side of problem is a nonsmoothfunctional with variable parameters. the existence of at least three positive solutions is establishedby using nonsmooth version of a three critical points theorem for discontinuous functions. our resul...

We are concerned with singular elliptic problems of the form −∆u ± p(d(x))g(u) = λf(x, u) + μ|∇u|a in Ω, where Ω is a smooth bounded domain in R , d(x) = dist(x, ∂Ω), λ > 0, μ ∈ R, 0 < a ≤ 2, and f, k are nonnegative and nondecreasing functions. We assume that p(d(x)) is a positive weight with possible singular behavior on the boundary of Ω and that the nonlinearity g is unbounded around the or...

We study the Cauchy problem ut = uxx + f (u) (t > 0, x ∈ R1), u(0, x) = u0(x) (x ∈ R1), where f (u) is a locally Lipschitz continuous function satisfying f (0) = 0. We show that any nonnegative bounded solution with compactly supported initial data converges to a stationary solution as t → ∞. Moreover, the limit is either a constant or a symmetrically decreasing stationary solution. We also con...

Based on an orientation-dependent dislocation line energy, a solution to the acoustic nonlinearity parameter is obtained for pure and mixed dislocations in anisotropic crystals. The solution is validated by comparison with molecular dynamic simulations. Parametric studies using this new solution show that (i) elastic anisotropy can significantly change the nonlinear behavior of dislocations inc...

The asymptotic behavior of global solutions of nonlinear evolution equations, in particular convergence to a certain equilibrium as time tends to infinity has become one of the main concerns in the field of nonlinear evolution equations since 1980s. For one space dimension case, significant progresses have been made (see e.g., [6]). However, the situation in higher space dimension case is much ...

We establish that every nonconstant bounded radial solution u of −∆u = f(u) in all of R is unstable if n ≤ 10. The result applies to every C nonlinearity f satisfying a generic nondegeneracy condition. In particular, it applies to every analytic and every power-like nonlinearity. We also give an example of a nonconstant bounded radial solution u which is stable for every n ≥ 11, and where f is ...

For a given nonzero bounded linear operator A on a Banach space X, we show that if A or A∗ has an eigenvalue then, except when the dimension of X is equal to two and the trace of A is zero, there exists a bounded linear operator B on X such that (i) AB + BA is of rank one, and (ii) I + f (A)B is invertible for every function f analytic in a neighborhood of the spectrum of A. This result was mot...