An applicable area of research that relates to PDE is inverse problems, for example the Calderon problem (see e.g. http://www.math.washington.edu/~gunther/publications/Papers/calderoniprevised.pdf ).

Applications include medical imaging (electrical impedance tomography), discovering oil under the sea, and finding cracks in concrete blocks.

Fields of mathematics related to inverse problems include PDE, microlocal analysis, integral transforms and Bayesian statistics. There is also numerical analysis.

Medical imaging and imaging concrete blocks are under current research at least in universities of Helsinki and eastern Finland, respectively.