3 Semilinear equations with absorption 19 3.1 The Marcinkiewicz spaces approach . . . . . . . . . . . . . . . . . . . . . . 21 3.2 Admissible measures and the ∆2-condition . . . . . . . . . . . . . . . . . . . 26 3.3 The duality method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.3.1 Bessel capacities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.3.2 Sharp...

We develop sufficient optimality conditions for a Moreau-Yosida regularized optimal control problem governed by a semilinear elliptic PDE with pointwise constraints on the state and the control. We make use of the equivalence of a setting of Moreau-Yosida regularization to a special setting of the virtual control concept, for which standard second order sufficient conditions have been shown. Mo...

3 Semilinear equations with absorption 19 3.1 The Marcinkiewicz spaces approach . . . . . . . . . . . . . . . . . . . . . . 20 3.2 Admissible measures and the ∆2-condition . . . . . . . . . . . . . . . . . . . 26 3.3 The duality method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.3.1 Bessel capacities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.3.2 Sharp...

We discuss regularity issues for minimizers of three nonlinear elliptic problems. They concern minimal cones, minimizing harmonic maps into a hemisphere, and radial local minimizers of semilinear elliptic equations. We describe the strong analogies among the three regularity theories. They all use a method originated in a paper of J. Simons on the area minimizing properties of cones.

We study boundary blow-up solutions of semilinear elliptic equations Lu = up + with p > 1, or Lu = e with a > 0, where L is a second order elliptic operator with measurable coefficients. Several uniqueness theorems and an existence theorem are obtained.

In this article, an optimal control problem subject to a semilinear elliptic equation and mixed control-state constraints is investigated. The problem data depends on certain parameters. Under an assumption of seperation of the active sets and a second-order sufficient optimality condition, Bouliganddifferentiability (B-differentiability) of the solutions with respect to the parameter is establ...

We consider the Dirichlet problem for semilinear elliptic equations on a smooth bounded domain Ω. We assume that Ω is symmetric about a hyperplane H and convex in the direction orthogonal to H. Employing Serrin’s result on an overdetermined problem, we show that any nonzero nonnegative solution is necessarily strictly positive. One can thus apply a well-known result of Gidas, Ni and Nirenberg t...

In this paper, we study a postprocessing procedure for improving accuracy of the finite volume element approximations of semilinear parabolic problems. The procedure amounts to solve a source problem on a coarser grid and then solve a linear elliptic problem on a finer grid after the time evolution is finished. We derive error estimates in the L and H norms for the standard finite volume elemen...

The numerical computation of solitary waves to semilinear elliptic equations in innnite cylinders is investigated. Rather than solving on the innnite cylinder, the equation is approximated by a boundary-value problem on a nite cylinder. Convergence and stability results for this algorithm are given. In addition, it is shown that Galerkin approximations can be used to calculate solitary waves fo...

An abstract framework is given to establish the existence and compute the Morse index of spike layer solutions of singularly perturbed semilinear elliptic equations. A nonlinear Lyapunov–Schmidt scheme is used to reduce the problem to one on a normally hyperbolic manifold, and the related linearized problem is also analyzed using this reduction. As an application, we show the existence of a mul...