Boundary singularities of solutions to elliptic viscous Hamilton–Jacobi equations

We study the boundary value problem with measures for (E1) −Δu+g(|∇u|)=0 in a bounded domain Ω in RN, satisfying (E2) u=μ on ∂Ω and prove that if is nondecreasing (E1)–(E2) can be solved with any positive bounded measure. When g(r)⩾rq 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)=rq with 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)=rq with qc⩽q<2 we prove that a necessary condition for solvability is that μ must be absolutely continuous with respect to the Bessel capacity . We also characterize boundary removable sets for moderate and sigma-moderate solutions.