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PDE & Potential Theory Group

The development of a "potential analysis" group in the department of mathematics at the Royal Institute of Technology (KTH) wasn't really the result of planning, it just evolved. The starting point was the discovery of "quadrature domains" (q.d.) which (in one incarnation) were first encountered by Dov Aharonov and Harold S. Shapiro in work on variational problems for conformal maps in the early seventies. The late Bo Kjellberg was enthusiastic about these ideas and encouraged H.S. Shapiro to pursue them with graduate students, and this advice, which was followed, seems to have borne fruit.

An interesting feature of q.d. is that they sit at a crossroads of nice ideas from complex analysis, potential theory, partial differential equations and (as was later discovered by Mihai Putinar) moment problems, and operator theory in Hilbert space. On the one hand, a q.d. is characterized as a uniform lamina having very remarkable "gravitational" problems: it attracts bodies far away just as if it were a finite collection of point masses. On the other hand, it is characterized by the solvability of a certain overdetermined Cauchy problem (for the Laplace operator), so the subject at once led (especially in its multivariable generalizations) to new problems involving Newtonian gravitation (especially so-called inverse problems) and overdetermined p.d.e. In the planar case the p.d.e. part was encapsulated in a "Schwarz function", originally envisaged for the elaboration of reflection principles w.r.t. analytic boundaries.

The spade work in sorting out all this was largely done in doctorate theses by Shapiro's students, especially (named in chronological order) Carina Ullemar, Bjoern Gustafsson, Gunnar Johnsson, Henrik Shahgholian and Peter Ebenfelt. One can also name here Lavi Karp, who began his research in Stockholm but finished his doctoral studies in Israel, with Aharonov, and who has maintained close contacts with our group. Furthermore, an expanding international constellation of mathematicians has pursued these studies, often in collaboration with our group. Their ranks include Boris Sternin and V. Shatalov (Moscow), Makoto Sakai (Tokyo), Dmitri Khavinson (Arkansas), Dov Aharonov (Haifa), Alexander Solynin (St. Petersburg), Mihai Putinar (Santa Barbara), John McCarthy (St. Louis), Liming Yang (Hawaii), Dmitry Yakubovich (St. Petersburg) , and Daoxing Xia (Nashville).

At the present time we have a small but active group at KTH, which is very well connected internationally. The fields of initial focus have spread further with the years and encompass variational inequalities, complementarity problems, free and moving boundary problems, ele Shaw flows, geometric measure theory, symmetrization, .... Stan Richardson (Edinburgh), Darren Crowdy (London), Rrobert Millar (Vancouver), David Armitage (Belfast) , Stephen Gardiner (Dublin), Mark Agranovsky (Israel).

Henrik Shahgholian's home page

Updated on 26 January, 2007.