Boyan Sirakov

Nonexistence of Positive Supersolutions of Elliptic Equations via the Maximum Principle

We introduce a new method for proving the nonexistence of positive supersolutions of elliptic inequalities in unbounded domains of ℝ n . The simplicity and robustness of our maximum principle-based argument provides for its applicability to many elliptic inequalities and systems, including quasilinear operators such as the p-Laplacian, and nondivergence form fully nonlinear operators such as Bellman-Isaacs operators. Our method gives new and optimal results in terms of the nonlinear functions appearing in the inequalities, and applies to inequalities holding in the whole space as well as exterior domains and cone-like domains.

Existence Results for Nonproper Elliptic Equations Involving the Pucci Operator

We study the equation in general smooth bounded domain Ω, and show it possesses nontrivial solutions provided that: f is sublinear, or f is superlinear and the equation admits a priori bounds. The existence result in the superlinear case is based on a new Liouville type theorem for − ℳλ,Λ +(D 2 u) = u p in a half-space.

Symmetry for exterior elliptic problems and two conjectures in potential theory

Abstract In this paper we extend a classical result of Serrin to a class of elliptic problems Δu+f(u,|∇u|)=0 in exterior domains R N ⧹G (or Ω⧹G with Ω and G bounded). In case G is an union of a finite number of disjoint C2-domains Gi and u=ai>0, ∂u/∂n=αi⩽0 on ∂Gi, u→0 at infinity, we show that if a non-negative solution of such a problem exists, then G has only one component and it is a ball. As a consequence we establish two results in electrostatics and capillarity theory. We further obtain symmetry results for quasilinear elliptic equations in the exterior of a ball.

Boundary Regularity for Viscosity Solutions of Fully Nonlinear Elliptic Equations

We provide regularity results at the boundary for continuous viscosity solutions to nonconvex fully nonlinear uniformly elliptic equations and inequalities in Euclidian domains. We show that (i) any solution of two sided inequalities with Pucci extremal operators is C 1, α on the boundary; (ii) the solution of the Dirichlet problem for fully nonlinear uniformly elliptic equations is C 2, α on the boundary; (iii) corresponding asymptotic expansions hold. This is an extension to viscosity solutions of the classical Krylov estimates for smooth solutions.

Singular Solutions of Fully Nonlinear Elliptic Equations and Applications

We study the properties of solutions of fully nonlinear, positively homogeneous elliptic equations near boundary points of Lipschitz domains at which the solution may be singular. We show that these equations have two positive solutions in each cone of

Soliton solutions to systems of coupled Schrödinger equations of Hamiltonian type

{\mathbb{R}^n}

Resonance Phenomena for Second-Order Stochastic Control Equations

, and the solutions are unique in an appropriate sense. We introduce a new method for analyzing the behavior of solutions near certain Lipschitz boundary points, which permits us to classify isolated boundary singularities of solutions which are bounded from either above or below. We also obtain a sharp Phragmén–Lindelöf result as well as a principle of positive singularities in certain Lipschitz domains.

Nonuniqueness for the Dirichlet Problem for Fully Nonlinear Elliptic Operators and the Ambrosetti–Prodi Phenomenon

We show that a class of divergence-form elliptic problems with quadratic growth in the gradient and non-coercive zero order terms are solvable, under essentially optimal hypotheses on the coefficients in the equation. In addition, we prove that the solutions are in general not unique. The case where the zero order term has the opposite sign was already intensively studied and the uniqueness is the rule.