arXiv subjects

Layan El Hajj

Publications and source records attributed to Layan El Hajj.

6 recordsLinked to original sources

arXiv2026

Existence theory for non-variational systems with free boundaries

We study the existence of solutions for systems of both elliptic and parabolic partial differential equations with potentially singular right-hand sides and free boundaries, posed in general smooth domains. Our results are established within a framework of "meta-theorems." This approach hinges on specific strong properties of the operators and their solutions (in approximate smooth settings) to guarantee the existence of a limit as the approximation parameter tends to zero. The primary challenge lies in applying these meta-theorems to prototype cases, which requires verifying that the necessary strong properties hold. For our analysis, we focus on fully nonlinear and $p$-Laplacian operators and a mixing of these operators, in both elliptic and parabolic contexts. While we focus on these specific cases, the meta-theorems remain valid for any other operators that satisfy the required properties. Beyond the complex proofs of our meta-theorems, and their applications to specific operators, a major challenge is the technical handling of the $p$-parabolic case, which requires proving the regularity of solutions of the $p$-parabolic equation with singular or degenerate right-hand side--addressed in the Appendix--along with several (new) properties, which is missing in the literature.

math.AP↗

arXiv2022

Symmetry for a fully nonlinear free boundary problem with highly singular term

In this paper we prove radial symmetry for solutions to a free boundary problem with a singular right hand side, in both elliptic and parabolic regime. More exactly, in the unit ball $B_1$ we consider a solution to the fully nonlinear elliptic problem $$ \begin{cases} F(D^2u)=f(u)&\text{in }B_1 \cap \{u >0 \},\\ u=M&\text{on }\partial B_1,\\ 0\le u 0\}$, we cannot apply the well-known Serrin-type boundary point lemma. We circumvent this by an exact assumption on a first order expansion and the decay on the second order, along with an ad-hoc comparison principle. We treat equally the parabolic case of the problem, and state a corresponding result.

math.AP↗

arXiv2022

On the Bohr's inequality for stable mappings

We consider the class of \emph{stable} harmonic mappings $f=h+\overline{g}$ introduced by Martin, Hernandez, and the class of \emph{stable} logharmonic mappings $f=zh\overline{g}$ introduced by AbdulHadi, El-Hajj. We determine Bohr's radius for the classes of stable univalent harmonic mappings, stable convex harmonic mappings and stable univalent logharmonic mappings. We also consider improved and refined versions of Bohr's inequality and discuss the Bohr's Rogonsiski radius for these family of mappings.

math.CV↗

arXiv2020

On the univalence of polyanalytic functions

A continuous complex-valued function $F$ in a domain $D\subseteq\mathbf{C}$ is Poly-analytic of order $α$ if it satisfies $\partial^α_{\overline{z}}F=0.$ One can show that $F$ has the form $F(z)={\displaystyle\sum\limits_{0}^{n-1}}\overline{z}^{k}A_{k}(z)$, where each $A_k$ is an analytic function$.$ In this paper, we prove the existence of a Landau constant for Poly-analytic functions and the special Bi-analytic case. We also establish the Bohr's inequality for poly-analytic and bi-analytic functions which map $U$ into $U$. In addition, we give an estimate for the arclength over the class of poly-analytic mappings and consider the problem of minimizing moments of order $p$.

math.CV↗

arXiv2016

On the univalence of polyharmonic mappings

A 2p-times continuously differentiable complex valued function $f = u + iv$ in a simply connected domain is polyharmonic (or p-harmonic) if it satisfies the polyharmonic equation $Δ^pF = 0$ . Every polyharmonic mapping f can be written as $f(z) =\sum_{k}^{p} |z|^{2(p-1)}G_{p-k+1}(z)$ where each $G_{p-k+1}$ is harmonic. In this paper we investigate the univalence of polyharmonic mappings on linearly connected domains and the relation between univalence of f(z) and that of $G_p(z)$. The notions of stable univalence and logpolyharminc mappings are also considered.

math.CV↗

arXiv2016

On geometrical properties of logharmonic mappings

In this paper, we find the radius of the disk $Ω_{r}$ such that every starlike logharmonic mapping $f(z)$ of order $α,$ is starlike in $% |z|\leq r$ with respect to any point of $Ω_{r}.$ We also establish a relation between the set of starlike logharmonic mappings \ and the set of starlike logharmonic mappings of order alpha. Moreover, the radius of starlikeness and univalence for the set of close to starlike logharmonic mappings of order $α$ is determined.

math.CV↗