Session S02 - Diverse Aspects of Elliptic PDEs and Related Problems

Thursday, July 15, 17:30 ~ 18:00 UTC-3

Asymptotic results in magnetic Orlicz-Sobolev spaces

Ariel Salort

Universidad de Buenos Aires, Argentina   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

We consider a non-local and non-standard growth version of the so-called magnetic Laplacian. This operator is known as the magnetic fractional $g-$Laplacian $(-\Delta_g^A)^s$ and it is defined as the gradient of the non-local energy functional \[ I_{s,G}^A(u) := \iint_{\mathbb{R}^n\times\mathbb{R}^n} \left(G\left(|\Re(D_s^A u(x,y))|\right) + G\left(|\Im(D_s^A u(x,y))|\right)\right)\, \frac{dxdy}{|x-y|^n}, \] where $G$ is a Young function and $D_s^A u(x,y)$ is the magnetic Holder quotient of order $s\in(0,1)$ defined as \[ D_s^A u(x,y) := \frac{u(x)-e^{i(x-y) A\left(\frac{x+y}{2}\right)} u(y)}{|x-y|^s}. \] Here $A:\mathbb{R}^n \to \mathbb{R}^n$ is a vector potential and $i$ denotes the imaginary unit.

We introduce briefly the notion of magnetic spaces in the fractional Orlicz-Sobolev setting and we study suitable Maz'ya-Shaposhnikova and Bourgain-Brezis-Mironescu formulas in modular form.

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