Sesión Ecuaciones Diferenciales y AplicacionesA Generalized Nonlocal Quasilinear Problem in Orlicz Spaces
Analía Silva
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A fractional counterpart of the \((p,q)\)-Laplacian is given by \((-\Delta_p)^s u + (-\Delta_q)^s u\), where \[(-\Delta_p)^s u(x):=C(n,s)\,\mathrm{p.v.}\int_{\mathbb{R}^n} \frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{n+sp}}\, dy.\] Several generalizations of this operator have been proposed in the setting of Fractional Orlicz spaces; however, these constructions breaks down when the two terms are allowed to have different fractional orders. To cover this and more general situations, in this talk, for \(s_1,\dots, s_k\in (0,1)\), we introduce a vector-valued fractional gradient \(D^{\mathbf s}u=(D^{s_1}u,\dots,D^{s_k}u)\) where \(\mathbf s=(s_1,\dots,s_k)\) and study energies for a generalized \(N\)-function \[M:\mathbb R^k\to\mathbb R_+\quad\iint_{\mathbb{R}^{2n}} M(D^{\mathbf s}u)\, d\nu_n \quad \mbox{ where} \quad D^s u(x,y) = \frac{u(x)-u(y)}{|x-y|^s}\quad\mbox{ and }\quad d\nu_n = \frac{dxdy}{|x-y|^n}.\]
We then develop the corresponding fractional Orlicz–Sobolev spaces and establish classical results in this context, such as a Poincaré inequality and a Rellich–Kondrashov-type compactness theorem. Finally, we present some existence results for equations involving this new operator.
This is joint work with Julián Fernández Bonder (IC-UBA) and Laura Cuneo (IC-UBA).