In this paper, we consider a boundary-asymptotic value problem for second-order uniformly elliptic equations with possibly unbounded coefficients. Problems of this kind arise in stochastic control settings. By assuming a positive drift, and the finiteness of a hybrid norm of the source, we obtain the existence of solutions, vanishing at infinity, in any unbounded domain, including the whole n-dimensional space (entire solutions). We use a new ABP type estimate, independent on the size and the geometry of domains, due to positive drift, possibly unbounded, and viscosity methods to carry out a converging approximation process. Suitable limitations on the spatial oscillations of second-order terms allow to obtain strong and classical solutions.

Existence of entire solutions vanishing at infinity by positive drift

Vitolo, Antonio
2025

Abstract

In this paper, we consider a boundary-asymptotic value problem for second-order uniformly elliptic equations with possibly unbounded coefficients. Problems of this kind arise in stochastic control settings. By assuming a positive drift, and the finiteness of a hybrid norm of the source, we obtain the existence of solutions, vanishing at infinity, in any unbounded domain, including the whole n-dimensional space (entire solutions). We use a new ABP type estimate, independent on the size and the geometry of domains, due to positive drift, possibly unbounded, and viscosity methods to carry out a converging approximation process. Suitable limitations on the spatial oscillations of second-order terms allow to obtain strong and classical solutions.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11386/4911140
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