Dirichlet Problem For Three Class Of Ornstein-Uhlenbeck Operators

In recent years,the boundary value problems of Ornstein-Uhlenbeck operators and Leray-Lious operators have received more and more attention from mathematicians.At present,there have been many studies on Ornstein-Uhlenbeck operators and Leray-Lious operators,which play a very important role in the fields of medicine,biology,chemistry,environmental protection,agriculture,astronomy and even economics.This paper mainly studies the existence and uniqueness of weak solutions and the continuity of weak solutions for the Dirichlet problem of three class of Ornstein-Uhlenbeck operators.Firstly,the existence of weak solutions of a class of linear Ornstein-Uhlenbeck operators for the Dirichlet problem is studied.By proving that it satisfies the three conditions of the Lax-Milgram theorem,it is concluded that the weak solutions exist and are unique.On this basis,we further consider the existence and uniqueness of the solutions to the Dirichlet problem for a class of linear Ornstein-Uhlenbeck operators with zero-order terms.The Lax-Milgram theorem is used to prove that the approximation equation has a weak solution un in and then use the interaction between the zero-order term?n(x)un and the right-hand term fn to obtain the consistent modulus maximum estimate of un and the energy estimate of un.Finally,the existence and uniqueness of the weak solution are obtained by taking the limit.Secondly,the Ornstein-Uhlenbeck operator and the Leray-Lious operator are combined to study the existence,uniqueness and weak continuity of the weak solutions of the Dirichlet problem for a class of nonlinear Ornstein-Uhlenbeck operators.The Galerkin approximation method is used to construct the approximate solution equation,and then the energy estimate is obtained.Then,the monotonicity of the Leray-Lious operator and the Minty method are used to obtain the existence and uniqueness of the weak solution.Finally,the weak continuity of the weak solution is obtained by the Minty method.