This study focuses on nonlocal boundary value problems (BVP) for linear and nonlinear elliptic differential-operator equations (DOE) that are defined in Banachvalued function spaces. The considered domain is a region with varying bound and depends on a certain parameter. Some conditions that guarantee the maximal Lpregularity and Fredholmness of linear BVP, uniformly with respect to this parame...

In this chapter, we present some of the major results that have been achieved in the context of the DFG-SPP project “Adaptive Wavelet Frame Methods for Operator Equations: Sparse Grids, Vector-Valued Spaces and Applications to Nonlinear Inverse Problems”. This project has been concerned with (nonlinear) elliptic and parabolic operator equations on nontrivial domains as well as with related inve...

The dynamics of matrix coupling with an application to Krylov methods p. 14 High precision method for calculating the energy values of the hydrogen atom in a strong magnetic field p. 25 Splitting methods and their application to the abstract Cauchy problems p. 35 Finite difference approximation of an elliptic interface problem with variable coefficients p. 46 The finite element method for the N...

this study concerns the existence and multiplicity of positive weak solutions for a class ofsemilinear elliptic systems with nonlinear boundary conditions. our results is depending onthe local minimization method on the nehari manifold and some variational techniques. alsoby using mountain pass lemma, we establish the existence of at least one solution withpositive energy.

We study nonlinear noncoercive elliptic problems with measure data, proving first that the global estimates already known when the problem is coercive are still true for noncoercive problems. We then prove new estimates, on sets far from the support of the singular part of the right-hand side, in the energy space associated to the operator, which entails additional regularity results on the sol...

In this paper, we study the finite element method for constrained optimal control problems governed by nonlinear elliptic PDEs. Instead of the standard error estimates under L2or H1norm, we apply the goal-oriented error estimates in order to avoid the difficulties which are generated by the nonsmoothness of the problem. We derive the a priori error estimates of the goal function, and the error ...

Much has been written about various obstacle problems in the context of variational inequalities. In particular, if the obstacle is smooth enough and if the coefficients of associated elliptic operator satisfy appropriate conditions, then the solution of the obstacle problem has continuous first derivatives. For a general class of obstacle problems, we show here that this regularity is attained...

We consider a class of second order elliptic problems in a domain of RN , N > 2, ε-periodically perforated by holes of size r(ε) , with r(ε)/ε → 0 as ε → 0. A nonlinear Robin-type condition is prescribed on the boundary of some holes while on the boundary of the others as well as on the external boundary of the domain, a Dirichlet condition is imposed. We are interested in the asymptotic behavi...

We consider nonlinear elliptic problems involving a nonlocal operator: the square root of the Laplacian in a bounded domain with zero Dirichlet boundary conditions. For positive solutions to problems with power nonlinearities, we establish existence and regularity results, as well as a priori estimates of Gidas-Spruck type. In addition, among other results, we prove a symmetry theorem of Gidas-...

This paper gives some global and uniform convergence estimates for a class of subspace correction (based on space decomposition) iterative methods applied to some unconstrained convex optimization problems. Some multigrid and domain decomposition methods are also discussed as special examples for solving some nonlinear elliptic boundary value problems.