Timeline for Characterize where the Dirichlet Problem for the Laplacian is always solvable
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17 events
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| May 26, 2011 at 1:48 | comment | added | Will Jagy | Yakov, excellent. Ahlfohrs says that the best result from Perron type arguments is this: The Dirichlet problem can be solved for any region whose complement is such that no component reduces to a point. He says that this provides a quick proof of Riemann mapping if taken as an axiom. The next section is multiply (but finitely) connected regions, concentric slit regions and so on. I don't think the original question by Linda has a nice answer. | |
| May 26, 2011 at 1:24 | comment | added | Yakov Shlapentokh-Rothman | @Will One can show that a simply connected two dimensional region admits a barrier function at each boundary point. I believe this is done in "Partial Differential Equations I: Basic Theory, Volume 1" by Michael Taylor in chapter 5 somewhere around the discussion of Riemann mapping for domains with a rough boundary. | |
| May 25, 2011 at 23:49 | comment | added | Will Jagy | However, it now appears that it is necessary to reconcile, within the same book by J. B. Conway, Corollary X.4.18 which asserts that "a simply connected region is a Dirichlet Region", and your original characterization with barriers. | |
| May 25, 2011 at 22:45 | comment | added | Will Jagy | Ahlfohrs gives the line-segment criterion as sufficient. I have no idea at this point whether there are (strong) solutions with arbitrary boundary data on the Koch snowflake. | |
| May 25, 2011 at 22:36 | comment | added | Linda Brown Westrick | @Nilima Yes, I would be very interested in a simple characterization that works only for the plane. Geometric conditions are often the right kind of simple. Now to be more specific about "simple": most of math can be formalized in Second Order Arithmetic; see en.wikipedia.org/wiki/Second-order_arithmetic for its syntax. The formula which says "G is a Dirichlet Region" has two set quantifiers (for all continuous $f$, there exists a harmonic $u$ such that...). Reduction to a single set quantifier would be useful, and an arithmetic characterization would be especially simple. | |
| May 25, 2011 at 22:06 | comment | added | Yakov Shlapentokh-Rothman | @Linda The Jost example is in three dimensions. I don't have Conway's book, but I imagine that his interest stems from the Riemann mapping theorem. Hence he probably only considers two dimensional regions. | |
| May 25, 2011 at 21:58 | comment | added | Linda Brown Westrick | @Will I think I found the theorem from McKernan you mentioned at math.ucsb.edu/~mckernan/Teaching/06-07/Spring/202C/l_5.pdf and it is a nice sufficient condition, but there is no claim there that it is necessary. (note: Theorem 5.5 is found in a set of McKernan's lecture notes, and no attribution is given there.) It would be interesting to see an example of a Dirichlet Region which did not meet this condition. | |
| May 25, 2011 at 21:45 | comment | added | Nilima Nigam | Linda, perhaps you could specify what you mean, then, by simpler. Are you looking for a geometric condition (similar to the cone condition)? And are you interested only in a characterization in the complex plane, or in $\mathbb{R}^n$? The Perron construction, and the characterization via regular points, is independent of number of dimensions. More specific characterizations are possible in the plane. | |
| May 25, 2011 at 21:39 | comment | added | Linda Brown Westrick | @Yakov There is a discrepancy between the cusp failure-example you are offering and Corollary X.4.18 of Conway's Functions of One Complex Variable I, which asserts that "a simply connected region is a Dirichlet Region". I am not familiar enough with potentials to be able to follow the example you referenced from Jost, but since I can see that the region is simply connected, perhaps there is some difference in the context between Jost and Conway. | |
| May 25, 2011 at 20:20 | comment | added | Linda Brown Westrick | @Nilima Thank you for your argument which shows why the boundary regularity condition characterizes the Dirichlet Regions. My question is asking something different, though. The boundary regularity condition, while necessary and sufficient, is a complicated condition. I am wondering if someone has found a simpler condition which also characterizes the Dirichlet Regions. | |
| May 25, 2011 at 18:29 | comment | added | Will Jagy | Thanks, Yakov. I haven't found it online yet. I remember Jost sitting in on a course by R. Schoen, 1980's. I see, the first edition is missing (stolen) from U.C.Berkeley math library (Evans Hall), but the second edition is available, QA377 .J66 2007 I may go photocopy pages 74-75 but leave it there for the OP to borrow. | |
| May 25, 2011 at 16:39 | comment | added | Yakov Shlapentokh-Rothman | @Will Jagy Check out "Partial Differential Equations" Second Edition by Jost pages 74-75, the end of the "alternating method of H.A. Schwarz" section. He gives an explicit example of a domain with an inward cusp where the solution to Dirichlet problem is not continuous up to the boundary. The relevant pages are available on google books. | |
| May 25, 2011 at 4:58 | comment | added | Andy Putman | nb : James McKernan is a very well-known algebraic geometer who has since moved to MIT. | |
| May 25, 2011 at 4:51 | comment | added | Will Jagy | Different page, someone named James McKernan at Santa Barbara (at the time), exterior cone suffices, his Theorem 5.5: The Dirichlet problem can be solved for any region $U$ such that each boundary point is the end point of a line segment whose other points are exterior to $U.$ So, a variant of the cardioid where the spine touching the origin oscillates would be an example, appropriate boundary data uncertain. math.ucsb.edu/~mckernan | |
| May 25, 2011 at 4:35 | comment | added | Will Jagy | I found a page by Boas that points out that the punctured disc in an example, take boundary data constant on the circle and a different constant at the origin. | |
| May 25, 2011 at 4:23 | comment | added | Will Jagy | Do you know a simple example of failure? I'm thinking a cardioid $r = 1 + \cos \theta$ with no external cone condition for one point, the origin. But I do not recall if it is exterior or interior cone that is relevant, nor a function to go with the shape. | |
| May 25, 2011 at 4:14 | history | answered | Nilima Nigam | CC BY-SA 3.0 |