We study semilinear elliptic equations on general bounded domains with concave semipositone nonlinearities. We prove the existence of the maximal solutions, and describe the global bifurcation diagrams. When a parameter is small, we obtain the exact global bifurcation diagram. We also discuss the related symmetry breaking bifurcation when the domains have certain symmetries.

We show that in a large class of semilinear elliptic binary optimal control problems, the point-wise states are submodular functions in the control variables almost everywhere. Moreover, we discuss how to use this result in order to design global optimization algorithms for such problems. To our knowledge, this is the rst time submodularity is investigated in the context of optimal control.

We consider semilinear elliptic problems of the form ∆u+g(u) = f(x) with Neumann boundary conditions or ∆u + λ1u + g(u) = f(x) with Dirichlet boundary conditions, and we derive conditions on g and f under which an upper bound on the number of solutions can be obtained.

Abstract. Semilinear elliptic optimal control problems involving the L1 norm of the control in the objective are considered. A priori finite element error estimates for piecewise linear discretizations for the control and the state are proved. These are obtained by a new technique based on an appropriate discretization of the objective function. Numerical experiments confirm the convergence rates.

Consider a nontrivial solution to a semilinear elliptic system of first order with smooth coefficients defined over an n-dimensional manifold. Assume the operator has the strong unique continuation property. We show that the zero set of the solution is contained in a countable union of smooth (n − 2)dimensional submanifolds. Hence it is countably (n − 2)-rectifiable and its Hausdorff dimension ...

Abstract A characterization of a semilinear elliptic partial differential equation (PDE) on bounded domain in R n is given terms an infinite-dimensional dynamical system. The system the space boundary data for PDE. This novel approach to problems that enables use systems tools studying corresponding ill-posed, meaning solutions do not exist forwards or backwards time generic initial data. We of...