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We investigate the effect of domain perturbation on the behavior of mild solutions for a class of semilinear stochastic partial differential equations subject to the Dirichlet boundary condition. Under some assumptions, we obtain an estimate for the mild solutions under changes of the domain.

Domain perturbation, or sometimes referred to as “perturbation of the boundary,” for boundary value problems is a special topic in perturbation problems. The motivation to study domain perturbation comes from various sources, which include shape optimization, solution structure of nonlinear problems, and numerical analysis. The main characteristic of domain perturbation is that the operators and the nonlinear terms live in different spaces, which leads to the solutions of partial differential equations living in different spaces. The fundamental question of domain perturbation is to consider how solutions behave upon varying domains. However, when we only consider the case of smooth perturbation of the domain, we could perform a change of variables to convert the perturbation problem into a fixed domain problem which is only perturbation of the coefficients. In this case, domain perturbation becomes back to a standard perturbation problem; in turn we may apply standard techniques such as the implicit function theorem, the Lyapunov-Schmidt method, and the transversality theorem to study it. Nevertheless, difficulties arrive when the change of variables and other standard tools do not work (see [

There are lots of papers concerning this topic [

Notice that all of works as we mentioned above are under the condition of Mosco convergence which describes the domain perturbation. For Dirichlet problems, it is worth pointing out that the condition of Mosco convergence conditions is equivalent to the strong convergence of resolvent operators (see [

Under the condition of the operator norm convergence of resolvent operators, the author of [

This paper is organized as follows: In Section

Throughout this paper, the letter

Let

We consider the stochastic equation as follows:

We adopt the following assumptions throughout this paper.

There exists a constant

Now we introduce the definition of mild solution to (

An

Let

Suppose the condition

In this section, we consider the following perturbation problem of (

Note the solutions value in different function spaces

To derive the solution of (

For

We assume that the nonlinear terms

For initial value of

By hypothesis

If

The result implies the upper semicontinuity of the spectrum; that is, if

Let the condition

By the relationship of spectrum and resolvent set, we have that hypothesis

Let

For every

If

As we all know, the relationship between resolvent operator and semigroup is denoted by

Let

By (

Let

For

Now we state and prove our main result as the following.

Suppose

From (

Denote

In this paper, we only consider the case in which Wiener process is scalar type; this result can not apply to the case of cylindrical Wiener processes. Note that if we concern the case of cylindrical Wiener processes, which relate to time and space, under perturbation of domain, cylindrical Wiener process is also perturbed which makes the situation more and more complicated.

The authors declare that they have no competing interests.

The authors would like to thank Professor Jinqiao Duan for helpful discussions and comments. This work was supported by NSFs of China (nos. 11271013 and 11526196) and the Fundamental Research Funds for the Central Universities (HUST: 2014TS066).