Continuous dependence estimates for viscosity solutions of fully nonlinear degenerate elliptic equations.
Continuous dependence on function parameters for superlinear Dirichlet problems
We discuss the existence of solutions for a certain generalization of the membrane equation and their continuous dependence on function parameters. We apply variational methods and consider the PDE as the Euler-Lagrange equation for a certain integral functional, which is not necessarily convex and coercive. As a consequence of the duality theory we obtain variational principles for our problem and some numerical results concerning approximation of solutions.
Convergence and partial regularity for weak solutions of some nonlinear elliptic equation : the supercritical case
Convergence of Cell Based Finite Volume Discretizations for Problems of Control in the Conduction Coefficients
We present a convergence analysis of a cell-based finite volume (FV) discretization scheme applied to a problem of control in the coefficients of a generalized Laplace equation modelling, for example, a steady state heat conduction. Such problems arise in applications dealing with geometric optimal design, in particular shape and topology optimization, and are most often solved numerically utilizing a finite element approach. Within the FV framework for control in the coefficients problems ...
Convergence of Cell Based Finite Volume Discretizations for Problems of Control in the Conduction Coefficients
We present a convergence analysis of a cell-based finite volume (FV) discretization scheme applied to a problem of control in the coefficients of a generalized Laplace equation modelling, for example, a steady state heat conduction. Such problems arise in applications dealing with geometric optimal design, in particular shape and topology optimization, and are most often solved numerically utilizing a finite element approach. Within the FV framework for control in the coefficients problems ...
Convergence of some adaptive FEM-BEM coupling for elliptic but possibly nonlinear interface problems
We consider the symmetric FEM-BEM coupling for the numerical solution of a (nonlinear) interface problem for the 2D Laplacian. We introduce some new a posteriori error estimators based on the (h − h/2)-error estimation strategy. In particular, these include the approximation error for the boundary data, which allows to work with discrete boundary integral operators only. Using the concept of estimator reduction, we prove that the proposed adaptive...
Convergence of some adaptive FEM-BEM coupling for elliptic but possibly nonlinear interface problems
We consider the symmetric FEM-BEM coupling for the numerical solution of a (nonlinear) interface problem for the 2D Laplacian. We introduce some new a posteriori error estimators based on the (h − h/2)-error estimation strategy. In particular, these include the approximation error for the boundary data, which allows to work with discrete boundary integral operators only. Using the concept of estimator reduction, we prove that the proposed adaptive...
Convergent algorithms suitable for the solution of the semiconductor device equations
In this paper, two algorithms are proposed to solve systems of algebraic equations generated by a discretization procedure of the weak formulation of boundary value problems for systems of nonlinear elliptic equations. The first algorithm, Newton-CG-MG, is suitable for systems with gradient mappings, while the second, Newton-CE-MG, can be applied to more general systems. Convergence theorems are proved and application to the semiconductor device modelling is described.
Convex integration with constraints and applications to phase transitions and partial differential equations
We study solutions of first order partial differential relations , where is a Lipschitz map and is a bounded set in matrices, and extend Gromov’s theory of convex integration in two ways. First, we allow for additional constraints on the minors of and second we replace Gromov’s −convex hull by the (functional) rank-one convex hull. The latter can be much larger than the former and this has important consequences for the existence of ‘wild’ solutions to elliptic systems. Our work was originally...
Convex symmetrization and applications
Convexity of level sets for solutions to nonlinear elliptic problems in convex rings.
Correction to my paper: “Existence theorems for operator equations and nonlinear elliptic boundary-value problems” [Comment. Math. Univ. Carolinae 14 (1973), 27-46]
Correction to the paper: local regularity for the solutions of the p-Laplacian on the Heisenberg group. The case
Counterexample to the convexity of level sets of solutions to the mean curvature equation
The convexity of level sets of solutions to the mean curvature equation is a long standing open problem. In this paper we give a counterexample to it.
Counterexample to the regularity of weak solution of elliptic systems
Counter-Examples to Boundary Estimates for Two-Dimensional Elliptic Systems.
Counterexamples to Some Conjectures on the Number of Solutions on Nonlinear Equations.
Critical groups computations on a class of Sobolev Banach spaces via Morse index
Critical Neumann problem for nonlinear elliptic systems in exterior domains.
Critical Neumann problem with competing Hardy potentials.