Monotonicity methods for white noise driven quasi-linear SPDEs. (English) Zbl 0722.60061

Buckdahn, R.; Pardoux, E.

Diffusion processes and related problems in analysis, Vol. I: Diffusions in analysis and geometry, Proc. Int. Conf., Evanston/IL (USA) 1989, Prog. Probab. 22, 219-233 (1990).

[For the entire collection see Zbl 0716.00011.]
The authors establish existence and uniqueness for nonlinear elliptic SPDEs driven by white noise \(\dot W,\) \[ -\Delta u(x)+f(u(x))=\dot W(x),\quad x\in D, \] where D is an open bounded subset of \(R^ k\) \((k=1,2,3)\) and f is an increasing function. The approach consists in generalizing the use of monotonicity by Lions to solve certain classes of deterministic nonlinear PDEs. By the same approach, in the second part of the paper nonlinear parabolic SPDEs \[ \frac{\partial u}{\partial t}(t,x)-\frac{\partial^ 2u}{\partial x^ 2}(t,x)+f(u)(t,x)=\frac{\partial^ 2W}{\partial t\partial x}(t,x)+g(t,x),\quad t\geq 0,\quad x\in [0,1], \] with space-time noise \(\partial^ 2W/\partial t\partial x\) and \(f=f_ 1+f_ 2\), \(f_ 1\) is increasing and \(f_ 2\) is Lipschitz, are studied.

Reviewer: R.Buckdahn (Berlin)

MSC:

60H15

Stochastic partial differential equations (aspects of stochastic analysis)

Keywords:

elliptic stochastic partial differential equations; parabolic stochastic partial differential equations; existence and uniqueness; monotonicity

Citations:

Zbl 0716.00011