No abstract

In this paper, in order to solve an elliptic partial differential equation with a nonlinear boundary condition for multiple solutions, the authors combine a minimax approach with a boundary integral-boundary element method, and identify a subspace and its special expression so that all numerical computation and analysis can be carried out more efficiently based on information of functions only on the boundary. Some mathematical justification of the new approach is established. An efficient and reliable local minimax-boundary element method is developed to numerically search for solutions. Details on implementation of the algorithm are also addressed. The existence and multiplicity of solutions to the problem are established under certain regular assumptions. Some conditions related to convergence of the algorithm and instability of solutions found by the algorithm are verified. To illustrated the method, numerical multiple solutions to some examples on domains with different geometry are displayed with their profile and contour plots.

We prove the existence of nondecreasing sequences of positive eigenvalues of the homogeneous degenerate quasilinear eigenvalue problem − div(a(x)|▽u|

Abstract. We prove an interpolation type inequality between C α , L ∞ and L p spaces and use it to establish the local Hölder continuity of the inverse of the p-Laplace operator:, for any f and g in a bounded set in L ∞ (Ω).

The present paper is devoted to the boundary formulation for anisotropic bodies subjected to a distribution of initial strains. Concerning both interior and exterior problems, the results are featured by the complete regularization of all the derived expressions: the integral representation of the displacement gradients and the stresses as well as the ordinary or derivative integral equations. The formulation also includes the case of piecewise regular boundary with edges or corners. Furthermore, explicit expressions in terms of physical meaningful boundary quantities such as the stresses related to the normal vector and the tangent derivatives of the displacements are given, which is particularly useful for further numerical implementations.