Elliptic boundary value problems with Gaussian white noise loads. (English) Zbl 1408.60051
Summary: Linear second order elliptic boundary value problems (BVP) on bounded Lipschitz domains are studied in the case of Gaussian white noise loads. The challenging cases of Neumann and Robin BVPs are considered.
The main obstacle for usual variational methods is the irregularity of the load. In particular, the Neumann boundary values are not well-defined.
In this work, the BVP is formulated by replacing the continuity of boundary trace mappings with measurability. Instead of variational methods alone, the novel BVP derives also from Cameron-Martin space techniques.
The new BVP returns the study of irregular white noise to the study of \(L^2\)-loads.
The main obstacle for usual variational methods is the irregularity of the load. In particular, the Neumann boundary values are not well-defined.
In this work, the BVP is formulated by replacing the continuity of boundary trace mappings with measurability. Instead of variational methods alone, the novel BVP derives also from Cameron-Martin space techniques.
The new BVP returns the study of irregular white noise to the study of \(L^2\)-loads.
MSC:
| 60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |
| 60H35 | Computational methods for stochastic equations (aspects of stochastic analysis) |
| 35R60 | PDEs with randomness, stochastic partial differential equations |
| 35J25 | Boundary value problems for second-order elliptic equations |
Keywords:
stochastic partial differential equations; boundary value problems; Gaussian measures; finite element methods; white noiseReferences:
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