I am looking for references on solutions of the $\overline{\partial}$ equation in the upper half-plane (with conditions on the boundary). But this subject is completely outside my area of expertise and therefore I am pretty much clueless. Unfortunately there is no standard way to key in $\overline{\partial}$ on Google, so my web search hasn't been successful. I would appreciate it if experts suggest some standard references on this subject.
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2$\begingroup$ Googling on "del bar equation" or "del bar operator" or "del bar problem" should give useful hits. $\endgroup$– paul garrettCommented Nov 8, 2013 at 18:16
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$\begingroup$ Please make the boundary conditions precise. $\endgroup$– Ben McKayCommented Mar 31, 2014 at 6:27
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3 Answers
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The standard reference is Hormander, An introduction to complex analysis in several variables. This is somewhat dense. An introduction for beginners (in one complex variable) is Berenstein, Carlos A.; Gay, Roger Complex variables. An introduction, Springer 1976.
answered Nov 8, 2013 at 17:46
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You may transform the problem into one on the unit disc. Then the theory in "The Neumann problem for the Cauchy Riemann complex" might apply. You may also try to solve the equation by hand on the unit disc using the Cauchy kernel (actually depending on your boundary conditions you might want to use ``the method of images" to find a variant of the Cauchy kernel that solves your boundary value problem).
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Unfortunately, usually you don't solve the $\bar \partial$ equation with boundary conditions. The reason is that if you find any solution, then adding a holomorphic function to it produces a new solution, and that can mess up any boundary conditions. If you impose boundary conditions, you could run into trouble. For this reason, people often don't solve $\bar \partial$ with boundary conditions. Instead, they either impose no conditions at all, or else ask for the solution to lie in some function space (when the data lies in some related function space). The simplest case is the $L^2$ case. That is explained in Hormander's book (mentioned above) but the discussion there is rather terse. There are excellent notes of Bo Berndtsson in a PCMI proceedings, which explain the $L^2$ theory much more digestibly.
In the $L^2$ case the problem is solved by a simple duality argument. There are dual forms involved, and THOSE FORMS satisfy a boundary condition which, in the case of the upper half plane, is just the so-called "Dirichlet condition": the forms must vanish on the boundary.
This is a beautiful and powerful theory, worth learning.
