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The Poincaré inequality ||u||p < C||Vu||p in a bounded domain holds, for instance, for compactly supported functions, for functions with mean value zero and for harmonic functions vanishing at a point. We show that it can be improved to ||u||p < C||6"Vu||p, where S is the distance to the boundary, and the positive exponent ß depends on the smoothness of the boundary.doi:10.2307/2047547 fatcat:vvgn2l76d5h4vo67ciaglyey3y