Rellich Type Inequalities with Weights in Plane Domains

© 2018, Pleiades Publishing, Ltd. We determine some special functionals as sharp constants in integral inequalities for test functions, defined on plane domains. First we prove a new one dimensional integral inequality. Also, we prove some generalizations of a classical Rellich result for two dimensional case, when there is an additional restriction for Fourier coefficients of the test functions. In addition, we examine a Rellich type inequality in plane domains with infinite Euclidean maximal modulus. As an application of our results we present a new simple proof of a remarkable theorem of P. Caldiroli and R. Musina from their paper “Rellich inequalities with weights”, published in Calc. Var. 45 (2012), 147–164