Differential Inclusions for Certain Differential Operators

Nesha, Nurun (2024) Differential Inclusions for Certain Differential Operators. PhD thesis, Indian Institute of Science Education and Research Kolkata.

Text (PhD thesis of Nurun Nesha (16IP021)) 16IP021.pdf - Submitted Version Restricted to Repository staff only Download (1MB)

Abstract

In this thesis, we study differential inclusions for various differential operators. In particular, we restrict our attention to the gradient, curl, and the symmetrized gradient operators. Out of these three, the differential inclusion involving the gradient operator is the most widely studied one, because its connection with other problems. We have identified the minimal dimension for the existence of a non-trivial solution and found a necessary and sufficient condition in this case. Next, we turn our attention to the curl operator. In this case, we are interested in the existence of a solution with the additional constraint of having a non-zero integral. In this case, we prove a few non-existence results. We also have found examples at dimension 2n − 3, where the differential inclusion admits a solution. Finally, in the case of the symmetrized gradient, we have found the minimal dimension to have a non-trivial solution.

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