Morteza Fotouhi

Sharif University of Technology | SHARIF · Department of Mathematical Science

This paper presents a mathematical framework for the prognosis of glioblastoma brain tumor growth on a patient-specific basis, employing a heterogeneous image-driven methodology. The approach utilizes a reaction-diffusion model to capture the diffusion and proliferation dynamics of tumor cell density, integrated with an inverse problem based on PDE...

Given is a bounded domain Ω ⊂ ℝ n {\Omega\subset\mathbb{R}^{n}} , and a vector-valued function defined on ∂ ⁡ Ω {\partial\Omega} (depicting temperature distributions from different sources), our objective is to study the mathematical model of a physical problem of enclosing ∂ ⁡ Ω {\partial\Omega} with a specific volume of insulating material to red...

We show that any minimizer of the well-known ACF functional (for the p-Laplacian) constitutes a viscosity solution. This allows us to establish a uniform flatness decay at the two-phase free boundary points to improve the flatness, which boils down to C1,η\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts}...

In this article, we consider a weakly coupled p p -Laplacian system of a Bernoulli-type free boundary problem, through minimization of a corresponding functional. We prove various properties of any local minimizer and the corresponding free boundary.

For a given constant λ > 0 and a bounded Lipschitz domain D ⊂ R n (n ≥ 2), we establish that almost-minimizers of the functional J(v; D) = D m i=1 |∇v i (x)| p + λχ {|v|>0} (x) dx, 1 < p < ∞, where v = (v 1 , · · · , v m), and m ∈ N, exhibit optimal Lipschitz continuity in compact sets of D. Furthermore , assuming p ≥ 2 and employing a distinctly d...

We study temperature distribution in a heat conducting problem, for a system of p-Laplace equation, giving rise to a free boundary.

In this paper, we study the classification of Lipschitz global solutions for a two-phase $p$-Laplace Bernoulli problem, subject to a mild assumption. Specifically, we focus on the scenario where the interior two-phase points of the global solution are non-empty. Our results show that the expected $C^{1,\eta}$ regularity holds in a suitable neighbor...

In this paper we are concerned with higher regularity properties of the elliptic system for \(0\le q<1\). We show analyticity of the regular part of the free boundary \(\partial \{|{\textbf{u}}|>0\}\), analyticity of \(|{\textbf{u}}|^{\frac{1-q}{2}} \) and \( \frac{{\textbf{u}}}{|{\textbf{u}}|}\) up to the regular part of the free boundary. Applyin...

We show that any minimizer of the well-known ACF functional (for the $p$-Laplacian) is a viscosity solution. This allows us to establish a uniform flatness decay at the two-phase free boundary points to improve the flatness, that boils down to $C^{1,\eta}$ regularity of the flat part of the free boundary. This result, in turn, is used to prove the...

In this paper we consider a weakly coupled $p$-Laplacian system of a Bernoulli type free boundary problem, through minimization of a corresponding functional. We prove various properties of any local minimizer and the corresponding free boundary.

In this work we address graph based semi-supervised learning using the theory of the spatial segregation of competitive systems. First, we define a discrete counterpart over connected graphs by using direct analogue of the corresponding competitive system. This model turns out doesn't have a unique solution as we expected. Nevertheless, we suggest...

In this paper we study the following parabolic system $$\begin{aligned} \Delta \mathbf{u }-\partial _t \mathbf{u }=|\mathbf{u }|^{q-1}\mathbf{u }\,\chi _{\{ |\mathbf{u }|>0 \}}, \qquad \mathbf{u }= (u^1, \cdots , u^m) \ , \end{aligned}$$ Δ u - ∂ t u = | u | q - 1 u χ { | u | > 0 } , u = ( u 1 , ⋯ , u m ) , with free boundary $$\partial \{|\mathbf{u...

This work is devoted to study a class of singular perturbed elliptic systems and their singular limit to a phase segregating system. We prove existence and uniqueness and study the asymptotic behavior of limiting problem as the interaction rate tends to infinity. The limiting problem is a free boundary problem such that at each point in the domain...

In this paper we study the following parabolic system \begin{equation*} \Delta \u -\partial_t \u =|\u|^{q-1}\u\,\chi_{\{ |\u|>0 \}}, \qquad \u = (u^1, \cdots , u^m) \ , \end{equation*} with free boundary $\partial \{|\u | >0\}$. For $0\leq q<1$, we prove optimal growth rate for solutions $\u $ to the above system near free boundary points, and show...

We study solutions and the free boundary ∂{|u|>0} of the sublinear systemΔu=λ+(x)|u+|q−1u+−λ−(x)|u−|q−1u−, from a regularity point of view. For λ±(x)>0 and Hölder, and 0<q<1, we apply the epiperimetric inequality approach and show C1,β-regularity for the free boundary at asymptotically flat points.

The purpose of this note is to present a “new” approach to the decay rate of the solutions to the no-sign obstacle problem from the free boundary, based on Weiss-monotonicity formula. In presenting the approach we have chosen to treat a problem which is not touched earlier in the existing literature. Although earlier techniques may still work for t...

We consider the homogenization of a Robin boundary value problem in a locally periodic perforated domain which is also time-dependent. We aim at justifying the homogenization limit, that we derive through asymptotic expansion technique. More exactly, we obtain the so-called corrector homogenization estimate that specifies the convergence rate. The...

We study existence, structure, uniqueness and regularity of solutions of the obstacle problem infu∈BVf(Ω)∫Ωϕ(x,Du),where BVf(Ω)={u∈BV(Rn):u≥ψinΩandu|∂Ω=f|∂Ω}, f∈W01,1(Rn), ψ is the obstacle, and ϕ(x,ξ) is a convex, continuous and homogeneous function of degree one with respect to the ξ variable. We show that every minimizer of this problem is also...

We study existence, structure, uniqueness and regularity of solutions of the obstacle problem \begin{equation*} \inf_{u\in BV_f(\Omega)}\int_{\mathbb{R}^n}\phi(x,Du), \end{equation*} where $BV_f(\Omega)=\{u\in BV(\Omega): u\geq \psi \text{ in }\Omega\text{ and } u|_{\partial \Omega}=f|_{\partial \Omega}\}$, $f \in W^{1,1}_0(\mathbb{R}^n)$, $\psi$ i...

Let Ω = R ² \ B(0 , 1) ¯ be the exterior of the closed unit ball. We prove the existence of extremal constant-sign solutions as well as sign-changing solutions of the following boundary value problem -Δu=a(x)f(u)inΩ,u=0on∂Ω=∂B(0,1),where the nonnegative coefficient a satisfies a certain integrability condition. We are looking for solutions in the s...

We study the regularity of solutions of the following semilinear problem $$\Delta u=-\lambda_+(x) (u^+)^{q}+\lambda_- (x) (u^-)^{q} \qquad \hbox{in } \ B_1,$$ where $B_1$ is the unit ball in $\bR^n$, $0<q<1 $ and $\lambda_\pm$ satisfy a H\"older continuity condition. Our main results concern local regularity analysis of solutions and their nodal s...

We study the seminilinear problem $$\Delta u=\lambda_+(x) (u^+)^{q-1}-\lambda_- (x) (u^-)^{q-1} \qquad \hbox{in } \ B_1,$$ from a regularity point of view for solutions and the free boundary $\partial\{\pm u>0\}$. Here $B_1$ is the unit ball, $1< q<2 $ and $\lambda_\pm$ are Lipschitz. Our main results concern local regularity analysis of solution...

The purpose of this paper is to address the question of well-posedness and spectral controllability of the wave equation perturbed by potential on networks which may contain unbounded potentials in the external edges. It has been shown before that in the absence of any potential, there exists an optimal time T∗ (which turns out to be simply twice t...

We present a universal model of brain tissue microstructure that dynamically links osmosis and diffusion with geometrical parameters of brain extracellular space (ECS). Our model robustly describes and predicts the nonlinear time dependency of tortuosity (?=D/D*) changes with very high precision in various media with uniform and nonuniform osmolari...

As long as the size of the embedded scatterer in comparison to the internal length scale of its surrounding elastic matrix is large, then the linear sampling method (LSM) and singular sources method (SSM) can be used in conjunction with classical theory of elasticity to reconstruct the size of the scatterer with reasonable accuracy. On the other ha...

We introduce an extension of the classical neural field equation where the dynamics of the synaptic kernel satisfies the standard Hebbian type of learning (synaptic plasticity). Here, a continuous network in which changes in the weight kernel occurs in a specified time window is considered. A novelty of this model is that it admits synaptic weight...

The purpose of this paper is to provide a full analysis of the null controllability problem for the one dimensional degenerate/singular parabolic equation u t − (a(x)u x) x − λ x β u = 0, (t, x) ∈ (0, T) × (0, 1), where the diffusion coefficient a(·) is degenerate at x = 0. Also the boundary conditions are considered to be Dirichlet or Neumann type...

The purpose of this paper is to provide a full analysis of the null controllability problem for the one dimensional degenerate/singular parabolic equation $ {u_t} - {\left( {a(x){u_x}} \right)_x} - \frac{\lambda }{{{x^\beta }}}u = 0 $ , (t, x) ∈ (0, T) × (0, 1), where the diffusion coefficient a(∙) is degenerate at x = 0. Also the boundary condit...

We introduce a modified-firing-rate model based on Hebbian-type changing synaptic connections. The existence and stability of solutions such as rest state, bumps, and traveling waves are shown for this type of model. Three types of kernels, namely exponential, Mexican hat, and periodic synaptic connections, are considered. In the former two cases,...

The singular sources method is given to detect the shape of a thin infinitely cylindrical obstacle from a knowledge of the TM-polarized scattered electromagnetic field in large distance. The basic idea is based on the singular behaviour of the scattered field of the incident point source on the cross-section of the cylinder. We assume that the scat...

We use the singular sources method to detect the shape of the obstacle in a mixed boundary value problem. The basic idea of the method is based on the singular behavior of the scattered field of the incident point-sources on the boundary of the obstacle. Moreover we take advantage of the scattered field estimate by the backprojection operator. Also...

In this paper, we use Conley index theory to examine the Poincare index of an isolated invariant set. We obtain some limiting conditions on a critical point of a planar vector field to be an isolated invariant set. As a result we show the existence of infinitely many homoclinic orbits for a critical point with the Poincare index greater than one.