arXiv subjects

Mohan Mallick

Publications and source records attributed to Mohan Mallick.

8 recordsLinked to original sources

arXiv2026

Existence and Multiplicity results for Weakly coupled system of Pucci's extremal operator

In this work, we investigate the existence of multiple positive solutions for a weakly coupled system of nonlinear elliptic equations governed by Pucci extremal operators. Specifically, we consider the system: \[ \begin{cases} -{M}_{\lambda_1,\Lambda_1}^+(D^2u_1) = \mu f_1(u_1, u_2, \dots, u_n), & \text{in } \Omega, \\ -{M}_{\lambda_2,\Lambda_2}^+(D^2u_2) = \mu f_2(u_1, u_2, \dots, u_n), & \text{in } \Omega, \vdots \\ -{M}_{\lambda_n,\Lambda_n}^+(D^2u_n) = \mu f_n(u_1, u_2, \dots, u_n), & \text{in } \Omega, \\ u_1 = u_2 = \dots = u_n = 0, & \text{on } \partial\Omega, \end{cases} \] where $ {M}_{\lambda,\Lambda}^+ $ represents the Pucci extremal operator, $ \Omega $ is a bounded domain in $ \mathbb{R}^N $ with smooth boundary, and the nonlinear functions $ f_i: [0, \infty)^n \to [0, \infty) $ belong to the $ C^{1,\alpha} $ class. Our main results establish the existence and multiplicity of solutions for sufficiently large values of the parameter $ \mu > 0 $. The analysis relies on the method of sub and supersolutions, in conjunction with fixed-point arguments and bifurcation techniques.

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arXiv2026

Shape Optimization for the Principal Eigenvalue of the Pucci Operator in Three Dimensions

We investigate shape optimization for the principal eigenvalue of the Pucci extremal operator \[ \left\{ \begin{aligned} -\mathcal{M}^+_{\lambda,\Lambda}(D^{2}u)&=\mu^{+}_{1}(\Omega)u &&\text{in }\Omega,\\ u &=0 &&\text{on }\partial\Omega, \end{aligned} \right. \] in dimension three. Since $\mathcal{M}^+_{\lambda,\Lambda}$ is fully nonlinear, in non-divergence form, and non-variational, classical symmetrization and rearrangement methods are not available. We introduce a three-dimensional family of double--pyramidal domains $\{\Omega^\omega_{\gamma,a}\}$ parametrized by an anisotropy factor $\gamma \in \left[\frac{1}{\sqrt{\omega}},\sqrt{\omega}\right]$ and an affine shear parameter $a\in(-\pi,\pi)$, under fixed ellipticity ratio $\omega=\Lambda/\lambda \ge 1$. Within this family and under a fixed-volume constraint, we prove that the volume-normalized principal eigenvalue is uniquely minimized at the symmetric unsheared configuration $(\gamma,a)=(1,0)$ among domains in the family $\{\Omega^\omega_{\gamma,a}\}$. The proof combines an explicit construction of positive eigenfunctions on seven patches with a lower bound under affine shear deformations. Using the homogeneity and orthogonal invariance of the Pucci operator, we identify an involutive symmetry $\gamma\mapsto \gamma^{-1}$ in the associated volume functional and establish strict monotonicity away from the self-dual point $\gamma=1$. In particular, for $\omega>1$, any nontrivial anisotropy or shear strictly increases the normalized principal eigenvalue. This reveals a genuinely three-dimensional rigidity mechanism for a fully nonlinear spectral problem and extends to dimension three the symmetry-minimization phenomenon previously known in the planar case.

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arXiv2026

Lane--Emden Systems with Singular Nonlinearities for the Fully Nonlinear Elliptic Operator

Consider \[ \begin{cases} F(D^2 u,Du,u,x) = u^{-p}v^{-q},~\text{in}~\Omega\\ F(D^2 v,Dv,v,x)=u^{-r}v^{-s},~~\text{in}~~\Omega\\ u,v>0~~\text{in}~~\Omega\\ u=v=0~\quad~\text{on}~~\partial\Omega, \end{cases} \] where $\Omega$ is an open connected subset of $\mathbb{R}^{N}$ and $p,s$ are two non-negative and $q,r$ are positive real numbers. This article discuses the conditions in terms of the relations among $p,q,r$ and $s$ which lead to existence, uniqueness and non-existence of positive solutions to the system. Furthermore, we also have studied some regularity properties of solution of the system. These results are inspired by the study of Lane-Emden system of equations as in \cite{busca2002liouville,ghergu2010lane}.

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arXiv2024

Exceptional Boundary Sets for Solutions of Fully Nonlinear Parabolic PDEs

This article investigates the exceptional set of the boundary for the following problem: \begin{equation*} \begin{aligned} -\frac{\partial u}{\partial t} + \mathcal{M}_{\lambda,\Lambda}^+(D^2u) + b(x,t)\cdot Du + c(x,t)u =0 \quad \rm{in} ~ \Omega_{T}, \end{aligned} \end{equation*} We provide a sufficient condition on the exceptional set in terms of the bound of the Hausdorff measure of this boundary portion. This condition ensures that even if the boundary values are not nonnegative on this portion, the supersolution remains nonnegative.

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arXiv2024

Regularity for Fully Nonlinear Elliptic Equations with Natural Growth in Gradient and Singular Nonlinearity

In this article we consider the following boundary value problem \begin{equation*}\label{abs} \left\{ \begin{aligned} F(x,u,Du,D^{2}u)+c(x)u+ p(x)u^{-\alpha}&=0~\text{in}~\Omega\\ u&=0~~\text{on}~~\partial\Omega, \end{aligned} \right. \end{equation*} where $\Omega$ is a bounded and $C^{2}$ smooth domain in $\mathbb{R}^N$ and $F$ has superlinear growth in gradient and $c(c)<-c_{0}$ for some positive constant $c_{0}.$ Here, we studies the boundary behaviour of the solutions to above equation and establishes the global regularity result similar to one established in [12,16] with linear growth in gradient.

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arXiv2024

Multiplicity results for fully nonlinear elliptic equations with natural gradient growth

In this paper, we prove a theorem concerning the existence of three solutions for the following boundary value problem: \begin{equation*} -\mathcal{M}_{\lambda,\Lambda}^+(D^2u)-\Gamma|Du|^2=f(u)~~~\text{in}\ \Omega, u=0~~~\text{on}\ \partial\Omega, \end{equation*} where $f:[0,\infty]\to[0,\infty]$ is a $C^{\alpha}$ function and $\Omega$ denotes a bounded, smooth domain in $\mathbb{R}^N$. By constructing two ordered pairs of sub and supersolutions for a specific class of $f$ exhibiting sublinear growth, we further establish the existence of three positive solutions to the aforementioned boundary value problem.

math.AP↗

arXiv2024

Optimal harvesting for a logistic model with grazing

We consider semi-linear elliptic equations of the following form: \begin{equation*} \left\{ \begin{aligned} -\Delta u &= \lambda[u-\dfrac{u^2}{K}-c \dfrac{u^2}{1+u^2}-h(x) u]=:\lambda f_h(u), \quad && x \in \Omega, \frac{\partial u}{\partial \eta}&+qu = 0, \quad && x\in\partial\Omega, \end{aligned} \right. \end{equation*} where, $h\in U=\{h\in L^2(\Omega): 0\leq h(x)\leq H\}.$ We prove the existence and uniqueness of the positive solution for large $\lambda.$ Further, we establish the existence of an optimal control $h\in U$ that maximizes the functional $J(h)=\int_{\Omega}h(x)u_h(x)~\rm{d}x-\int_{\Omega}(B_1+B_2 h(x))h(x)~\rm{d}x$ over $U$, where $u_h$ is the unique positive solution of the above problem associated with $h$, $B_1>0$ is the cost per unit effort when the level of effort is low and $B_2>0$ represents the rate at which the cost rises as more labor is employed. Finally, we provide a unique optimality system.

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arXiv2022

On a shape derivative formula for the Robin $p$-Laplace eigenvalue

We obtain shape derivative formulae for the first eigenvalue of the Robin $p$-Laplace operator. This result is used to study the variation of the first eigenvalue with respect to perturbations of the domain. In particular, we prove that for large values of the boundary parameter, the first eigenvalue is monotonic with respect to domain inclusion for smooth domains.

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