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We investigate the doubling inequality and nodal sets for the solutions ofbi-Laplace equations. A polynomial upper bound for the nodal sets of solutionsand their gradient is obtained based on the recent development of nodal setsfor Laplace eigenfunctions by Logunov. In addition, we derive an implicit upperbound for the nodal sets of solutions. We show two types of doublinginequalities for the solutions of bi-Laplace equations. As a consequence, therate of vanishing is given for the solutions.

Doubling inequality and nodal sets for solutions of bi-Laplace equations