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The Neumann problem on unbounded domains of \(\mathbb R^d\) and stochastic variational inequalities. (English) Zbl 1130.35137

The following elliptic problem with Neumann boundary conditions is considered \[ \begin{gathered} \lambda\varphi-{1\over 2}\Delta\varphi+ \langle F(\chi), \nabla\varphi(\chi)\rangle= f\quad\text{in }\mathring K,\\ {\partial\varphi\over\partial n}= 0\quad\text{on }\partial K,\end{gathered}\tag{1} \] where \(\lambda> 0\), \(K\) is a closed convex subset of \(\mathbb{R}^d\), generally unbounded, with \(C^2\) boundary \(\partial K\) and nonempty interior \(\mathring K\), and \({\partial\varphi\over\partial n}\) is the outward normal derivative to \(\partial K\). For the sake of simplicity, we shall assume that \(d> 1\) and that \(0\in\mathring K\).
Moreover, the vector field \(F: \mathbb{R}^d\to \mathbb{R}^d\) is quasi-monotone, coercive and has polynomial growth.
The main novelty of this very interesting paper is that this problem is studied in \(L^2(\mathbb{R}^d,\nu)\) where \(\nu\) is not Lebesgue measure, but an invariant measure for the stochastic process solving the corresponding stochastic variational inequality \[ dX(t)+ F(X(t))\,dt+ \partial I_K(X(t))\,dt\ni dW(t).\tag{2} \] It is also proved that the semigroup generated by the elliptic operator in (1) coincides with the transition semigroup of the solution to (2). This result is the first of its kind for such singular coefficients and Neumann boundary conditions.

MSC:

35R60 PDEs with randomness, stochastic partial differential equations
35J85 Unilateral problems; variational inequalities (elliptic type) (MSC2000)
60J35 Transition functions, generators and resolvents
35J25 Boundary value problems for second-order elliptic equations
47D06 One-parameter semigroups and linear evolution equations
47H06 Nonlinear accretive operators, dissipative operators, etc.
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