We consider a class of semilinear elliptic problems in twoand three-dimensional domains with conical points. We introduce Sobolev spaces with detached asymptotics generated by the asymptotical behaviour of solutions of corresponding linearized problems near conical boundary points. We show that the corresponding nonlinear operator acting between these spaces is Fréchet differentiable. Applying ...

In this paper we consider optimal control of stochastic semilinear equations with linearly increasing drift and cylindrical noise. We show existence and uniqueness (up to an additive constant) of solutions to the stationary Hamilton-Jacobi equation associated with the cost functional given by the asymptotic average per unit time cost. As a consequence we nd the optimizing controls given in the ...

In this article, we generalize the main result obtained by Bahaj in [1]. Also our proof is shorter than the original proof.

We want to explain the crucial difference between the second order (m = 1) and higher order (m > 1) case. If m = 1 a very satisfactory result is known. Indeed, the sign condition (2) alone is sufficient to ensure classical solvability of the Dirichlet problem (1). There are two very strong devices in the theory of second order elliptic equations, which make the proof of the needed a-priori maxi...