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We present a general linear response description of membrane adhesion atrough or chemically structured surfaces. Our method accounts for non-local Vander Waals effects and contains the more approximate (and local) Deryaginapproach in a simple limit. Specializing to supported membranes we consider theeffects of substrate structure on the membrane adhesion energy andconfiguration. Adhesion is usually less favorable for rough substrates and themembrane shape tends to follow that of the surface contours. Chemicalpatterning, however, favors adhesion with the membrane configuration being outof phase with the surface structure. Finally, considering a surface indentedwith `V'-shaped trenches, we show that our approach is in fair agreement withan exact numerical solution.Comment: 21 pages, 7 ps figures, submitted to PR

Supported membranes on chemically structured and rough surfaces