The Moser-Trudinger embedding has been generalized by Adimurthi and Sandeep to the following weighted version: if ⊂ R 2 is bounded, α > 0 and β ∈ [0, 2) are such thatWe prove that the supremum is attained, generalizing a well-known result by Flucher, who has proved the case β = 0.

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We demonstrate a novel technique to fabricate mechanically tunable slippery surfaces using one-dimensional (anisotropic) elastic wrinkles. Such wrinkles show tunable topography (amplitude) on the application of mechanical strain. Following Nepenthes pitcher plants, lubricating fluid infused solid surfaces show excellent slippery behavior for test liquid drops. Therefore, combining the above two, that is, infusing suitable lubricating fluid on elastic wrinkles, would enable us to fabricate mechanically tunable slippery surfaces. Completely stretched (flat) wrinkles have uniform coating of lubricating fluid, whereas completely relaxed (full amplitude) wrinkles have most of the lubricating oil in the wrinkle grooves. Therefore, water drops on completely stretched surface show excellent slippery behavior, whereas on completely relaxed surface they show reduced slippery behavior. Therefore, continuous variation of wrinkle stretching provides reversibly tunable slippery behavior on such a system. Because the wrinkles are one-dimensional, they show anisotropic tunability of slippery behavior depending upon whether test liquid drops slip parallel or perpendicular to the wrinkles.

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The mobility of liquid drops on lubricant-infused slippery surfaces depends strongly on various system parameters, for example, surface energy and roughness of the underlying solid surface and surface tension and viscosity of the test and the lubricating fluids. Here, we investigate lubricant-coated slippery surfaces fabricated on smooth hydrophobic solid surfaces and examine the influence of thickness and viscosity of the lubricating oil on the velocity of aqueous drops. We also investigate the effect of surface tension of the test liquid using a binary mixture of water and ethanol, on the apparent contact angle, which further affects their slip velocity. A theoretical model, based on various dissipative forces acting in different regions of the lubricating oil and a test drop, is also presented, which elucidates the dependence of drop velocity on lubricating oil viscosity and base radius of drops of test liquids.