arXiv subjects

Irshaad Ahmed

Publications and source records attributed to Irshaad Ahmed.

4 recordsLinked to original sources

arXiv2024

Applications of Interpolation theory to the regularity of some equasilinear PDEs

We present some regularity results on the gradient of the weak or entropic-renormalized solution $u$ to the homogeneous Dirichlet problem for the quasilinear equations of the form \begin{equation*}\label{p-laplacian_eq} -{\rm div~}(|\nabla u|^{p-2}\nabla u)+V(x;u)=f, \end{equation*} where $Ω$ is a bounded smooth domain of $\mathbb R^n$, $V$ is a nonlinear potential and $f$ belongs to non-standard spaces like Lorentz-Zygmund spaces. Moreover, we collect some well-known and new results for identifying some interpolation spaces and enrich some contents with details.

math.AP↗

arXiv2024

A generalized version of Holmstedt's formula for the K-functional

Let $(A_0, A_1)$ be a compatible couple of quasi-normed spaces, and let $Φ_0$ and $Φ_1$ be two general parameters of $K$-interpolation method. We compute $K$-functional for the couple $((A_0,A_1)_{Φ_0}, (A_0, A_1)_{Φ_1})$ in terms of $K$-functional for the couple $(A_0, A_1)$.

math.FA↗

arXiv2023

Quasilinear P.D.Es, Interpolation spaces and Hölderian mappings

As in the work of Tartar ( Tartar L. Interpolation non linéaire et régularité, 9, Journal of Functional Analysis, (1972), 469-489) we developed here some new results on non linear interpolation of $α$-Hölderian mappings between normed spaces, namely, by studying the action of the mappings on $K$-functionals and between interpolation spaces with logarithm functors. We apply those results to obtain regularity results on the gradient of the solution to quasilinear equations of the form $$-div(\widehat a(\nabla u ))+V(u)=f, $$ whenever $V$ is a nonlinear potential, $f$ belongs to non standard spaces as Lorentz-Zygmund spaces. We show among other that the mapping $T: \ Tf=\nabla u$ is locally or globally $α$-Hölderian under suitable values of $α$ and adequate hypothesis on $V$ and $\widehat a.$

math.AP↗

arXiv2022

Holmstedt's formula for the $K$-functional: the limit case $θ_0=θ_1$

We consider $K$-interpolation spaces involving slowly varying functions, and derive necessary and sufficient conditions for a Holmstedt-type formula to be held in the limiting case $θ_0=θ_1\in\{0,1\}.$ We also study the case $θ_0=θ_1\in (0,1).$ Applications are given to Lorentz-Karamata spaces, generalized gamma spaces and Besov spaces.

math.FA↗