Boundary singularities of solutions to elliptic viscous Hamilton-Jacobi equations

Contents 1 Introduction 2 2 The Dirichlet problem and the boundary trace 7 2. Abstract We study the boundary value problem with measures for (E1) −∆u + g(|∇u|) = 0 in a bounded domain Ω in R N , satisfying (E2) u = µ on ∂Ω and prove that if g ∈ L 1 (1, ∞; t −(2N +1)/N dt) is nondecreasing (E1)-(E2) can be solved with any positive bounded measure. When g(r) ≥ r q with q > 1 we prove that any positive function satisfying (E1) admits a boundary trace which is an outer regular Borel measure, not necessarily bounded. When g(r) = r q with 1 < q < q c = N +1 N we prove the existence of a positive solution with a general outer regular Borel measure ν ≡ / ∞ as boundary trace and characterize the boundary isolated singularities of positive solutions. When g(r) = r q with q c ≤ q < 2 we prove that a necessary condition for solvability is that µ must be absolutely continuous with respect to the Bessel capacity C 2−q q ,q ′. We also characterize boundary removable sets for moderate and sigma-moderate solutions.