L^p -uniqueness for Dirichlet operators with singular potentials |
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Authors: | Vitali Liskevich Oleksiy Us |
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Institution: | (1) Institute for Applied Problems of Mechanics and Mathematics, 3b Naukova st., 79601 Lviv, Ukraine and Lviv National University, 1 Universytetska st., 79602 Lviv, Ukraine;(2) Lviv National University, 1 Universytetska st., 79602 Lviv, Ukraine |
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Abstract: | We study the problem of strong uniqueness in Lp for the Dirichlet operator perturbed by a singular complex-valued potential. First we construct the generator -Hp of a C0-semigroup in Lp, with Hp extending the restriction of the perturbed Dirichlet operator to the set of smooth functions. The corresponding sesquilinear form in L2 is not assumed to be sectorial. Then we reveal sufficient conditions on the logarithmic derivative # of the measure rdx \rho dx and the potential q which ensure that -Hp is the only extension of D+b·?-q \upharpoonrightC0¥ \Delta +\beta \cdot \nabla -q \upharpoonright_{C_0^{\infty}} which generates a C0-semigroup on Lp. The method of a priori estimates of solutions to corresponding differential equations is employed. |
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