scholarly journals Extension theory and Kreĭn-type resolvent formulas for nonsmooth boundary value problems

2014 ◽  
Vol 266 (7) ◽  
pp. 4037-4100 ◽  
Author(s):  
Helmut Abels ◽  
Gerd Grubb ◽  
Ian Geoffrey Wood
Keyword(s):  
Boundary Value Problems ◽  
Boundary Value ◽  
Extension Theory ◽  
Nonsmooth Boundary

scholarly journals Three-Field Modelling of Nonlinear Nonsmooth Boundary Value Problems and Stability of Differential Mixed Variational Inequalities

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
J. Gwinner
Keyword(s):  
Variational Inequalities ◽  
Saddle Point ◽  
Boundary Value Problems ◽  
Solid Mechanics ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Initial Boundary ◽  
Set Convergence ◽  
Nonsmooth Boundary ◽  
Mixed Variational Inequalities

The purpose of this paper is twofold. Firstly we consider nonlinear nonsmooth elliptic boundary value problems, and also related parabolic initial boundary value problems that model in a simplified way steady-state unilateral contact with Tresca friction in solid mechanics, respectively, stem from nonlinear transient heat conduction with unilateral boundary conditions. Here a recent duality approach, that augments the classical Babuška-Brezzi saddle point formulation for mixed variational problems to twofold saddle point formulations, is extended to the nonsmooth problems under consideration. This approach leads to variational inequalities of mixed form for three coupled fields as unknowns and to related differential mixed variational inequalities in the time-dependent case. Secondly we are concerned with the stability of the solution set of a general class of differential mixed variational inequalities. Here we present a novel upper set convergence result with respect to perturbations in the data, including perturbations of the associated nonlinear maps, the nonsmooth convex functionals, and the convex constraint set. We employ epiconvergence for the convergence of the functionals and Mosco convergence for set convergence. We impose weak convergence assumptions on the perturbed maps using the monotonicity method of Browder and Minty.

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