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In this paper we study a first extension of the theory of mild solutions for HJB equations in Hilbert spaces to the case when the domain is not the whole space. More precisely, we consider a half-space as domain, and a semilinear Hamilton-Jacobi-Bellman (HJB) equation. Our main goal is to establish the existence and the uniqueness of solutions to such HJB equations, that are continuously differentiable in the space variable. We also provide an application of our results to an exit time optimal control problem and we show that the corresponding value function is the unique solution to a semilinear HJB equation, possessing sufficient regularity to express the optimal control in feedback form. Finally, we give an illustrative example.

Let $X$ be a separable Banach space endowed with a non-degenerate centered Gaussian measure $\mu$. The associated Cameron-Martin space is denoted by $H$. Let $\nu=e^{-U}\mu$, where $e^{-U}$ is a sufficiently regular weight and $U:X\rightarrow\mathbb{R}$ is a convex and continuous function. In this paper we are interested in the $W^{2,2}$ regularity of the weak solutions of elliptic equations of the type \[\lambda u-L_\nu u=f,\] where $\lambda>0$, $f\in L^2(X,\nu)$ and $L_\nu$ is the self-adjoint operator associated with the quadratic form \[(\psi,\varphi)\mapsto \int_X\left\langle\nabla_H\psi,\nabla_H\varphi\right\rangle_Hd\nu\qquad\psi,\varphi\in W^{1,2}(X,\nu).\]

We study the elliptic equation $\lambda u-L^{\Omega}u=f$ in an open convex subset $\Omega$ of an infinite dimensional separable Banach space $X$ endowed with a centered non-degenerate Gaussian measure $\gamma$, where $L^\Omega$ is the Ornstein-Uhlenbeck operator. We prove that for $\lambda>0$ and $f\in L^2(\Omega,\gamma)$ the weak solution $u$ belongs to the Sobolev space $W^{2,2}(\Omega,\gamma)$. Moreover we prove that $u$ satisfies the Neumann boundary condition in the sense of traces at the boundary of $\Omega$. This is done by finite dimensional approximation.

We study the Ornstein-Uhlenbeck operator and the Ornstein-Uhlenbeck semigroup in an open convex subset of an infinite dimensional separable Banach space $X$. This is done by finite dimensional approximation. In particular we prove Logarithmic-Sobolev and Poincar\'e inequalities, and thanks to these inequalities we deduce the spectral properties of the Ornstein-Uhlenbeck operator.

In this paper we study a H\'enon-like equation (see equations (1) below), where the nonlinearity f(t) is not homogeneous (i.e., it is not a power). By minimization on the Nehari manifold, we prove that for large values of the parameter $\alpha$ there is a breaking of symmetry and non radial solutions appears. This holds for sub- and super-critical growth of the nonlinearity f.