English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Open

Presents the pdf analysis as premium feature

PDF Analysis

Insights

Presents the summarisation as premium feature

Summarisation

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

We develop a linear theory of very weak solutions for nonlocal eigenvalueproblems $\mathcal L u = \lambda u + f$ involving integro-differentialoperators posed in bounded domains with homogeneous Dirichlet exteriorcondition, with and without singular boundary data. We consider mild hypotheseson the Green's function and the standard eigenbasis of the operator. The mainexamples in mind are the fractional Laplacian operators.  Without singular boundary datum and when $\lambda$ is not an eigenvalue ofthe operator, we construct an $L^2$-projected theory of solutions, which weextend to the optimal space of data for the operator $\mathcal L$. We present aFredholm alternative as $\lambda$ tends to the eigenspace and characterise thepossible blow-up limit. The main new ingredient is the transfer oforthogonality to the test function.  We then extend the results to singular boundary data and study the so-calledlarge solutions, which blow up at the boundary. For that problem we show that,for any regular value $\lambda$, there exist "large eigenfunctions" that aresingular on the boundary and regular inside. We are also able to present aFredholm alternative in this setting, as $\lambda$ approaches the values of thespectrum.  We also obtain a maximum principle for weighted $L^1$ solutions when theoperator is $L^2$-positive. It yields a global blow-up phenomenon as the firsteigenvalue is approached from below.  Finally, we recover the classical Dirichlet problem as the fractionalexponent approaches one under mild assumptions on the Green's functions. Thus"large eigenfunctions" represent a purely nonlocal phenomenon.

Blow-up phenomena in nonlocal eigenvalue problems: when theories of $L^1$ and $L^2$ meet