In a paper I read, an elliptic boundary value problem on a bounded domain D x (0,T) is solved by first transforming it in a set of equations on half-spaces R^n and then applying partial Fourier transform. Eventually one single ODE is obtained. It is not explained how this transformation is done and it seems very mysterious to me. Can someone please give references for this approach? I am also interested if it is possible in general for a single evolution equation to be transformed in system of equations on half spaces. Thank you for all replies.
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1$\begingroup$ Could you give a reference to the paper in question? It would make it a lot easier to provide some sort of answer. $\endgroup$– Harald Hanche-OlsenCommented Nov 14, 2009 at 18:13
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$\begingroup$ I have to agree with @Harald. Does T represent a time parameter here? $\endgroup$– MLeviCommented Nov 14, 2009 at 21:43
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$\begingroup$ Mary Ann Horn:"Sharp trace regularity..", Journal of Math.Systems, Estimation and Control, vol.8, No2, 1998. Yes T is time variable. In which volume of Hörmander book is this explained? I didn't find anything in the first one. $\endgroup$– martintonCommented Nov 15, 2009 at 11:13
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The half spaces R^n are the inward-pointing halves of the tangent spaces to the points on the boundary. Starting with a local boundary value problem you end up with a model equation on the half space associated to each point on the boundary by "localizing" at the point. A solution to the elliptic boundary value problem would give you also a solution to each of these model problems. Conversely, you can construct a formal solution to the elliptic boundary value problem at the boundary by using these model problems. Then you can improve this formal solution to an `honest' solution by solving away the error; this is easier because you no longer have to worry about the boundary.
This system of ODEs is used to state the Lopatinski-Shapiro condition, and searching for this will yield many references. My favorite is Hormander's analysis of partial differential operators -- this topic in particular is best explained in the first edition.
