| Abstract: |
| This talk shall be about a class of elliptic PDEs involving the mixed operator
$$\mathcal{L}= -\Delta+(-\Delta)^{s},~ \text{for}~s \in (0, 1)$$
which are prescribed with mixed(Dirichlet-Neumann) boundary conditions. I shall discuss the corresponding eigenvalue problems and bifurcation type result(both from zero and infinity) for an asymptotically linear problem inclined with eigenvalue problems.
Firstly, I will provide a functional setup that embeds a variational structure to problems with mixed opeartor and mixed boundary conditions. Under this framework, our first main result was the existence of the first eigenvalue and corresponding positive eigenfunctions with their expected characteristics viz. principal eigenvalue, simple and strictly positive. We also establish boundedness and H\older regularity of eigenfunctions. We also show a strong maximum principle and a few other regularity results which are independent of interest.
Secondly, I will show some asymptotic behavior of the first eigenvalues with respect to the Dirichlet set, when Neumann sets dissipate and vice versa.
Lastly, I will discuss bifurcation type results, built upon the strong maximum principle and H\older regularity results, for an asymptotically linear problem inclined with the eigenvalue problems. |
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