Local vs. Nonlocal: interpolation and couplings

ACOSTA, GABRIEL

Montevideo

Workshop; NoLoCe Nonlocal and Local Coupled Equations; 2023

Universidad de la República

Weighted and fractional Sobolev spaces associated with seminorms of the type \[\int_\Omega \int_\Omega \frac{|f(x)-f(y)|^p}{|x-y|^{n+sp}} \, \delta^{\beta}(x,y) \, dy \,dx,\]where $\delta(x,y)=\min\{d(x), d(y)\}$, have shown to be useful for establishing regularity results and developing efficient numerical methods for nonlocal equations involving the fractional Laplace operator [Acosta, G., Borthagaray, J. P., A fractional Laplace equation: regularity of solutions and finite element approximations. SIAM J. Numer. Anal. (2017)].In the first part of this talk [Acosta G., Drelichman, I., Durán, R. G., Weighted fractional Sobolev spaces as interpolation spaces in bounded domains, to appear in Math. Nach.], we characterize the real interpolation space between $L^p$ and a weighted Sobolev space involving weights that are certain positive powers of the distance to the boundary. In particular,\[(L^p(\Omega), W^{1,p}(\Omega,1,d^{ p}))_{s,p},\]is characterized by means of the seminorm \[\int_\Omega {\int_{|x-y|