arXiv subjects

Hussien Abugirda

Publications and source records attributed to Hussien Abugirda.

5 recordsLinked to original sources

arXiv2026

An introduction to the vectorial Calculus of Variations in $\textbf{L}^\infty$ AND Aronsson PDE systems

In this expository article, which is partially based on the lecture notes of a short course delivered by the second appearing author, we introduce the subfield of the Calculus of Variations that is concerned with the study of vectorial variational problems for supremal functionals, with emphasis on the associated PDE systems arising as extremality conditions. The scalar theory has a rather short history, first arising in the work of G. Aronsson in the 1960s. The vectorial case is even more recent and first arose in work of the second appearing author in the early 2010s. The Calculus of Variations in $L^\infty$ is an all-important field for numerous applications, but it presents serious difficulties and requires new tools, different to those required for the classical case of integral functionals and Euler-Lagrange equations.

math.AP↗

arXiv2018

Rigidity and flatness of the image of certain classes of mappings having tangential Laplacian

In this paper we consider the PDE system of vanishing normal projection of the Laplacian for $C^2$ maps $u : \mathbb{R}^n \supseteq Ω\longrightarrow \mathbb{R}^N$: \[ [\![\mathrm{D} u]\!]^\bot Δu = 0 \ \, \text{ in }Ω. \] This system has discontinuous coefficients and geometrically expresses the fact that the Laplacian is a vector field tangential to the image of the mapping. It arises as a constituent component of the $p$-Laplace system for all $p\in [2,\infty]$. For $p=\infty$, the $\infty$-Laplace system is the archetypal equation describing extrema of supremal functionals in vectorial Calculus of Variations in $L^\infty$. Herein we show that the image of a solution $u$ is piecewise affine if either the rank of $\mathrm{D} u$ is equal to one or $n=2$ and $u$ has the additively separated form $u(x,y)=f(x)+g(y)$. As a consequence we obtain corresponding flatness results for the images of $p$-Harmonic maps, $p\in [2,\infty]$.

math.AP↗

arXiv2018

Phase separation of n dimensional infinity Harmonic mappings

Among other interesting results, in a recent paper, Katzourakis analysed the phenomenon of separation of the solutions to the infinity Laplace system to phases with qualitatively different behavior in the case of the 2 dimensional infinity Harmonic mappings. The solutions of the infinity Laplace system are called the infinity Harmonic mappings. In this paper we discuss an extension of Katzourakis result mentioned above to higher dimensions by studying the phase separation of n dimensional infinity Harmonic mappings.

math.AP↗

arXiv2016

Existence of 1D vectorial Absolute Minimisers in $L^\infty$ under minimal assumptions

We prove the existence of vectorial Absolute Minimisers in the sense of Aronsson to the supremal functional $E_\infty(u,Ω') = \|\mathscr{L}(\cdot,u,D u)\|_{L^\infty(Ω')}$, $Ω'\Subset Ω$, applied to $W^{1,\infty}$ maps $u:Ω\subseteq \mathbb{R}\longrightarrow \mathbb{R}^N$ with given boundary values. The assumptions on $\mathscr{L}($ are minimal, improving earlier existence results previously established by Barron-Jensen-Wang and by the second author.

math.AP↗

arXiv2016

On the Well-Posedness of Global Fully Nonlinear First Order Elliptic Systems

In the very recent paper [K1], the second author proved that for any $ f\in L^2(\mathbb{R}^n,\mathbb{R}^N)$, the fully nonlinear first order system $F(\cdot,\mathrm{D} u) =f$ is well posed in the so-called J.L. Lions space and moreover the unique strong solution $u:\mathbb{R}^n\longrightarrow \mathbb{R}^N$ to the problem satisfies a quantitative estimate. A central ingredient in the proof was the introduction of an appropriate notion of ellipticity for $F$ inspired by Campanato's classical work in the 2nd order case. Herein we extend the results of [K1] by introducing a new strictly weaker ellipticity condition and by proving well posedness in the same "energy" space.

math.AP↗