{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:00:21Z","timestamp":1787320821228,"version":"build-2736575974"},"reference-count":76,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Control Optim."],"published-print":{"date-parts":[[2015,1]]},"abstract":"<jats:p>Controlling droplet shape via surface tension has numerous technological applications, such as droplet lenses and lab-on-a-chip. This motivates a partial differential equation-constrained shape optimization approach for controlling the shape of droplets on flat substrates by controlling the surface tension of the substrate. We use shape differential calculus to derive an $L^2$ gradient flow approach to compute equilibrium shapes for sessile droplets on substrates. We then develop a gradient-based optimization method to find the substrate surface tension coefficient yielding an equilibrium droplet shape with a desired footprint (i.e., the liquid-solid interface has a desired shape). Moreover, we prove a sensitivity result with respect to the substrate surface tensions for the free boundary problem associated with the footprint. Numerical results are also presented to showcase the method.<\/jats:p>","DOI":"10.1137\/140979721","type":"journal-article","created":{"date-parts":[[2015,3,19]],"date-time":"2015-03-19T14:52:39Z","timestamp":1426776759000},"page":"771-799","source":"Crossref","is-referenced-by-count":12,"title":["Droplet Footprint Control"],"prefix":"10.1137","volume":"53","author":[{"given":"Antoine","family":"Laurain","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Shawn W.","family":"Walker","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2015,3,19]]},"reference":[{"key":"atypb1","unstructured":"A. Alarcon and R. Souam,\n                      Capillary Surfaces Inside Polyhedral Regions\n                      , arXiv:1401.6935, 2014."},{"key":"atypb2","doi-asserted-by":"crossref","unstructured":"H. W. Alt,\n                      \n                        Verzweigungspunkte von $H$-Fla\u0308chen.\n                        I\n                      \n                      , Math. 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Brochard-Wyart, and D. Que\u0301re\u0301,\n                      Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls, Waves\n                      , Springer-Verlag, New York, 2004.","DOI":"10.1007\/978-0-387-21656-0"},{"key":"atypb16","unstructured":"M. C. Delfour and J.P. Zole\u0301sio,\n                      Shapes and Geometries: Analysis, Differential Calculus, and Optimization\n                      , Adv. Des. Control 4, SIAM, Philadelphia, 2001."},{"key":"atypb17","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2006.10.046"},{"key":"atypb18","doi-asserted-by":"publisher","DOI":"10.1007\/BF01385643"},{"key":"atypb19","doi-asserted-by":"publisher","DOI":"10.4171\/IFB\/163"},{"key":"atypb20","unstructured":"A. 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