Singularities of Nonlinear Elliptic Systems

Crack singularities for general elliptic systems

We consider general homogeneous Agmon-Douglis-Nirenberg elliptic systems with constant coefficients complemented by the same set of boundary conditions on both sides of a crack in a two-dimensional domain. We prove that the singular functions expressed in polar coordinates (r; ) near the crack tip all have the form r 12+k'( ) with k 0 integer, with the possible exception of a finite number of s...

Conformally invariant fully nonlinear elliptic equations and isolated singularities

1 Introduction There has been much work on conformally invariant fully nonlinear elliptic equations and applications to geometry and topology. [10], and the references therein. In this and a companion paper [16] we address some analytical issues concerning these equations. For n ≥ 3, consider −∆u = n − 2 2 u n+2 n−2 , on R n .

Existence of Critical Elliptic Systems with Boundary Singularities

Lorentz Spaces and Nonlinear Elliptic Systems

In this paper we study the following system of semilinear elliptic equations:  −∆u = g(v) , in Ω, −∆v = f(u) , in Ω, u = 0 and v = 0 , on ∂Ω, where Ω is a bounded domain in R , and f, g ∈ C(R) are superlinear nonlinearities. The natural framework for such systems are Sobolev spaces, which give in most cases an adequate answer concerning the ”maximal growth” on f and g such that the proble...

Nonlinear Elliptic Systems and Mean Field Games ∗

We consider a class of quasilinear elliptic systems of PDEs consisting of N HamiltonJacobi-Bellman equations coupled with N divergence form equations, generalising to N > 1 populations the PDEs for stationary Mean-Field Games first proposed by Lasry and Lions. We provide a wide range of sufficient conditions for the existence of solutions to these systems: either the Hamiltonians are required t...