A large part of my research has been concerned with free boundary problems. These problems involve solving partial differential equations in domains whose shape has to be found as part of the solution. They occur in many physical and industrial applications. Examples include waves propagating at the interface between two fluids, bubbles rising in a fluid and growing tumors. Mathematically free boundary problems are challenging because of their nonlinearity. Over the years I have developed efficient and accurate boundary integral equation methods to solve free boundary problems occuring in fluid mechanics.

I have devoted special attention to the effect of surface tension and discovered classes of problems which have a continuum of solutions when surface tension is neglected but only a discrete set of solutions when surface tension is taken into account. Moreover I have shown that this discrete set reduces to a unique solution as the surface tension tends to zero. Some of my recent research is concerned with three dimensional free surface flows and the discovery of three dimensional solitary waves. I am also involved in studying the effects of electric fields on interfacial flows.