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Kaushik Bal

Publications and source records attributed to Kaushik Bal.

At least 19 recordsLinked to original sources
arXiv2026

Local Existence, Uniqueness, Regularity, and Global Behavior of Evolution Equations Involving Mixed Local and Nonlocal Operators

In this work, we address a parabolic problem featuring a potentially doubly nonlinear term, governed by a combination of local and nonlocal operators (see Problem P1 below). We first establish the local existence of weak energy solutions via a semidiscretization in time applied to an auxiliary evolution problem. The uniqueness of these solutions is subsequently obtained through a novel generalization of the classical inequality of Diaz and Saa, suitably adapted to the mixed local nonlocal setting. This generalization provides a new comparison principle and establishes the T-accretivity of a corresponding operator in L2. By employing this comparison principle, we construct suitable barrier functions that allow the global in time extension of solutions. Furthermore, we demonstrate the convergence of weak solutions to a nontrivial stationary state. Our approach relies on methods from the theory of contraction semigroups. It is noteworthy that these results are underpinned by a detailed analysis of the stationary problems associated with Problem P1, which also reveals several qualitative properties of the solutions.

math.AP↗
arXiv2025

On some Elliptic and Parabolic Problems Involving the Anisotropic $p(u)$-Laplacian

We investigate a class of elliptic and parabolic partial differential equations driven by p(u) laplacian. This dependence necessitates the use of variable exponent Sobolev spaces specifically tailored to the anisotropic framework. For the elliptic case, we establish the existence of a weak solution by employing the theory of pseudomonotone operators in conjunction with suitable approximation techniques. In the parabolic setting, the existence of a weak solution is obtained via a time discretization scheme and Schauder fixed-point theorem, supported by a priori estimates and compactness arguments.

math.AP↗
arXiv2025

Comparison principle for Singular Fractional $ g- $Laplacian Problems

In this paper, we establish a novel comparison principle of independent interest and prove the uniqueness of weak solutions within the local Orlicz--Sobolev space framework, for the following class of fractional elliptic problems: \begin{equation*} (-\Delta)^{s}_{g} u = f(x) u^{-\alpha} + k(x) u^{\beta}, \quad u > 0 \quad \text{in } \Omega; \quad u = 0 \quad \text{in } \mathbb{R}^{N} \setminus \Omega, \end{equation*} where \( \Omega \subset \mathbb{R}^{N} \) is a smooth bounded domain, \( \alpha > 0 \), and \( \beta > 0 \) satisfies a suitable upper bound. Here, \( (-\Delta)^{s}_{g} \) denotes the fractional \( g \)-Laplacian, with \( g \) being the derivative of a Young function \( G \). The function \( f \) is assumed to be nontrivial, while \( k \) is a positive function, and both \( f \) and \( k \) are assumed to lie in suitable Orlicz spaces. Our analysis relies on a refined variational approach that incorporates a \( G \)-fractional version of the D\'iaz--Saa inequality together with a \( G \)-fractional analogue of Picone's identity. These tools, which are of independent interest, also play a key role in the study of simplicity of eigenvalues, Sturmian-type comparison results, Hardy-type inequalities, and related topics.

math.AP↗
arXiv2024

Multiplicity of solutions to a class of degenerate elliptic equations in both sub-critical and critical cases

Given a smooth, bounded domain $\Omega\subset\mathbb{R}^N$, we establish the existence of two non-trivial, non-negative solutions to the semilinear degenerate elliptic equation \begin{align*} \left. \begin{array}{l} -\Delta_\lambda u=\mu g(z)|u|^{r-1}u+h(z)|u|^{s-1}u \;\text{in}\; \Omega u\in H^{1,\lambda}_0(\Omega) \end{array}\right\} \end{align*} where $\Delta_\lambda=\Delta_x+|x|^{2\lambda}\Delta_y$ denotes the Grushin Laplacian Operator, $z=(x,y)\in\Omega$, $N=n+m;\, n,\, m\geq 1$, $\lambda>0$, $0\leq r<1<s<2^*_\lambda-1$ and $\mu$ is a positive parameter. The functions $g$ and $h$ may change sign and $2^*_\lambda=\frac{2Q}{Q-2}$ is the critical Sobolev exponent associated with the homogeneous dimension $Q=n+(1+\lambda)m$ of $\Delta_\lambda$. In the critical case $s=2^*_\lambda-1$, we further show that the problem admits at least two non-trivial, non-negative solutions under the additional assumptions $g\geq 0$ and $h\equiv 1$.

math.AP↗
arXiv2024

Regularity results for a class of mixed local and nonlocal singular problems involving distance function

We investigate the following mixed local and nonlocal quasilinear equation with singularity given by \begin{eqnarray*} \begin{split} -\Delta_pu+(-\Delta)_q^s u&=\frac{f(x)}{u^{\delta}}\text { in } \Omega, \\u&>0 \text{ in } \Omega,\\u&=0 \text { in }\mathbb{R}^n \backslash \Omega; \end{split} \end{eqnarray*} where, \begin{equation*} (-\Delta )_q^s u(x):= c_{n,s}\operatorname{P.V.}\int_{\mathbb{R}^n}\frac{|u(x)-u(y)|^{q-2}(u(x)-u(y))}{|x-y|^{n+sq}} d y, \end{equation*} with $\Omega$ being a bounded domain in $\mathbb{R}^{n}$ with $C^2$ boundary, $1 0$ and $f\in L^\infty_{\mathrm{loc}}(\Omega)$ is a non-negative function which behaves like $\mathbf{dist(x,\partial \Omega)^{-\beta}}$, $\beta\geq 0$ near $\partial \Omega$. We start by proving several H\"older and gradient H\"older regularity results for a more general class of quasilinear operators when $\delta=0$. Using the regularity results we deduce existence, uniqueness and H\"older regularity of a weak solution of the singular problem in $W_{\mathrm{loc}}^{1,p}(\Omega)$ and its behavior near $\partial \Omega$ albeit with different exponents depending on $\beta+\delta$. Boundedness and H\"older regularity result to the singular equation with critical exponent were also discussed.

math.AP↗
arXiv2024

Multiplicity of solutions for mixed local-nonlocal elliptic equations with singular nonlinearity

We will prove multiplicity results for the mixed local-nonlocal elliptic equation of the form \begin{eqnarray} \begin{split} -Δ_pu+(-Δ)_p^s u&=\fracλ{u^γ}+u^r \text { in } Ω, \\u&>0 \text{ in } Ω,\\u&=0 \text { in }\mathbb{R}^n \backslash Ω; \end{split} \end{eqnarray} where \begin{equation*} (-Δ)_p^s u(x)= c_{n,s}\operatorname{P.V.}\int_{\mathbb{R}^n}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{n+sp}} d y, \end{equation*} and $-Δ_p$ is the usual $p$-Laplace operator. Under the assumptions that $Ω$ is a bounded domain in $\mathbb{R}^{n}$ with regular enough boundary, $p>1$, $n> p$, $s\in(0,1)$, $λ>0$ and $r\in(p-1,p^*-1)$ where $p^*$ is the critical Sobolev exponent, we will show there exist at least two weak solutions to our problem for $0<γ<1$ and some certain values of $λ$. Further, for every $γ>0$, assuming strict convexity of $Ω$, for $p=2$ and $s\in(0,1/2)$, we will show the existence of at least two positive weak solutions to the problem, for small values of $λ$, extending the result of \cite{garaingeometric}. Here $c_{n,s}$ is a suitable normalization constant, and $\operatorname{P.V.}$ stands for Cauchy Principal Value.

math.AP↗
arXiv2024

Global stability and optimal control in a single-strain dengue model with fractional-order transmission and recovery process

The current manuscript introduce a single-strain dengue model developed from stochastic processes incorporating fractional order transmission and recovery. The fractional derivative has been introduced within the context of transmission and recovery process, displaying characteristics similar to tempered fractional ($TF$) derivatives. It has been established that under certain condition, a function's $TF$ derivatives are proportional to the function itself. Applying the following observation, we examined stability of several steady-state solutions, such as disease-free and endemic states, in light of this newly formulated model, using the reproduction number (R_0). In addition, the precise range of epidemiological parameters for the fractional order model was determined by calibrating weekly registered dengue incidence in the San Juan municipality of Puerto Rico, from April 9, 2010, to April 2, 2011. We performed a global sensitivity analysis method to measure the influence of key model parameters (along with the fractional-order coefficient) on total dengue cases and the basic reproduction number (R_0) using a Monte Carlo-based partial rank correlation coefficient (PRCC). Moreover, we formulated a fractional-order model with fractional control to asses the effectiveness of different interventions, such as reduction the recruitment rate of mosquito breeding, controlling adult vector, and providing individual protection. Also, we established the existence of a solution for the fractional-order optimal control problem. Finally, the numerical experiment illustrates that, policymakers should place importance on the fractional order transmission and recovery parameters that capture the underline mechanisms of disease along with reducing the spread of dengue cases, carried out through the implementation of two vector controls.

math.DS↗
arXiv2024

On a mixed local-nonlocal evolution equation with singular nonlinearity

We will prove several existence and regularity results for the mixed local-nonlocal parabolic equation of the form \begin{eqnarray} \begin{split} u_t-Δu+(-Δ)^s u&=\frac{f(x,t)}{u^{γ(x,t)}} \text { in } Ω_T:=Ω\times(0, T), \\ u&=0 \text { in }(\mathbb{R}^n \backslash Ω) \times(0, T), \\ u(x, 0)&=u_0(x) \text { in } Ω; \end{split} \end{eqnarray} where \begin{equation*} (-Δ)^s u= c_{n,s}\operatorname{P.V.}\int_{\mathbb{R}^n}\frac{u(x,t)-u(y,t)}{|x-y|^{n+2s}} d y. \end{equation*} Under the assumptions that $γ$ is a positive continuous function on $\overlineΩ_T$ and $Ω$ is a bounded domain %of class $\mathcal{C}^{1,1}$ with Lipschitz boundary in $\mathbb{R}^{n}$, $n> 2$, $s\in(0,1)$, $0<T<+\infty$, $f\geq 0$, $u_0\geq 0$, $f$ and $u_0$ belongs to suitable Lebesgue spaces. Here $c_{n,s}$ is a suitable normalization constant, and $\operatorname{P.V.}$ stands for Cauchy Principal Value.

math.AP↗
arXiv2023

Ground state solutions for quasilinear Schrodinger type equation involving anisotropic p-laplacian

This paper is concerned with the existence of a nonnegative ground state solution of the following quasilinear Schrödinger equation \begin{equation*} \begin{split} -Δ_{H,p}u+V(x)|u|^{p-2}u-Δ_{H,p}(|u|^{2α}) |u|^{2α-2}u=λ|u|^{q-1}u \text{ in }\;R^n;\; u\in W^{1,p}(\;R^n)\cap L^\infty(\;R^N) \end{split} \end{equation*} where $N\geq2$; $(α,p)\in D_N=\{(x,y)\in \;R^2 : 2xy\geq y+1,\; y\geq2x,\; y 0$ is a parameter. The operator $Δ_{H,p}$ is the reversible Finsler p-Laplacian operator with the function $H$ being the Minkowski norm on $\;R^N$. Under certain conditions on $V$, we establish the existence of a non-trivial non-negative bounded ground state solution of the above equation.

math.AP↗
arXiv2023

On the singular problem involving $g$-Laplacian

In this paper, we show that the existence of a positive weak solution to the equation $(-Δ_g)^s u=f u^{-q(x)}\;\mbox{in}\; Ω,$ where $Ω$ is a smooth bounded domain in $R^N$, $q\in C^1(\overlineΩ)$, and $(-Δ_g)^s$ is the fractional $g$-Laplacian with $g$ is the antiderivative of a Young function and $f$ in suitable Orlicz space subjected to zero Dirichlet condition. This includes the mixed fractional $(p,q)-$Laplacian as a special case. The solution so obtained is also shown to be locally Hölder continuous.

math.AP↗
arXiv2023

Magnetic fractional Poincaré inequality in punctured domains

We study Poincaré-Wirtinger type inequalities in the framework of magnetic fractional Sobolev spaces. In the local case, Lieb-Seiringer-Yngvason [E. Lieb, R. Seiringer, and J. Yngvason, Poincaré inequalities in punctured domains, Ann. of Math., 2003] showed that, if a bounded domain $Ω$ is the union of two disjoint sets $Γ$ and $Λ$, then the $L^p$-norm of a function calculated on $Ω$ is dominated by the sum of magnetic seminorms of the function, calculated on $Γ$ and $Λ$ separately. We show that the straightforward generalisation of their result to nonlocal setup does not hold true in general. We provide an alternative formulation of the problem for the nonlocal case. As an auxiliary result, we also show that the set of eigenvalues of the magnetic fractional Laplacian is discrete.

math.FA↗
arXiv2023

Semilinear degenerate elliptic equation in the presence of singular nonlinearity

Given $Ω(\subseteq\;R^{1+m})$, a smooth bounded domain and a nonnegative measurable function $f$ defined on $Ω$ with suitable summability. In this paper, we will study the existence and regularity of solutions to the quasilinear degenerate elliptic equation with a singular nonlinearity given by: \begin{align} -Δ_λu&=\frac{f}{u^ν} \text{ in }Ω\nonumber &u>0 \text{ in } Ω\nonumber &u=0 \text{ on } \partialΩ\nonumber \end{align} where the operator $Δ_λ$ is given by $$Δ_λ{u}=u_{xx}+|x|^{2λ}Δ_y{u};\,(x,y)\in \;R\times\;R^m $$ is known as the Grushin operator.

math.AP↗
arXiv2023

Existence of radial solution for a quasilinear equation with singular nonlinearity

We prove that the equation \begin{eqnarray*} -Δ_p u =λ\Big( \frac{1} {u^δ} + u^q + f(u)\Big)\;\text{ in } \, B_R(0) u =0 \,\text{ on} \; \partial B_R(0), \quad u>0 \text{ in } \, B_R(0) \end{eqnarray*} admits a weak radially symmetric solution for $λ>0$ sufficiently small, $0<δ<1$ and $p-1<q<p^{*}-1$. We achieve this by combining a blow-up argument and a Liouville type theorem to obtain a priori estimates for the regularized problem. Using a variant of a theorem due to Rabinowitz we derive the solution for the regularized problem and then pass to the limit.

math.AP↗
arXiv2021

Hardy and Poincaré inequalities in fractional Orlicz-Sobolev spaces

We provide sufficient conditions for boundary Hardy inequality to hold in bounded Lipschitz domains, complement of a point (the so-called point Hardy inequality), domain above the graph of a Lipschitz function, the complement of a bounded Lipschitz domain in fractional Orlicz-Sobolev setting. As a consequence, we get sufficient conditions for regional fractional Orlicz Poincaré inequality in bounded Lipschitz domains. Necessary conditions for fractional Orlicz Hardy and regional fractional Orlicz Poincaré inequalities are also given for bounded Lipschitz domains. Various sufficient conditions on open sets are provided for fractional Orlicz Poincaré inequality and regional fractional Orlicz Poincaré inequality to hold.

math.AP↗
arXiv2021

Weighted anisotropic Sobolev inequality with extremal and associated singular problems

For a given Finsler-Minkowski norm $\mathcal{F}$ in $\mathbb{R}^N$ and a bounded smooth domain $Ω\subset\mathbb{R}^N$ $\big(N\geq 2\big)$, we establish the following weighted anisotropic Sobolev inequality $$ S\left(\int_Ω|u|^q f\,dx\right)^\frac{1}{q}\leq\left(\int_Ω\mathcal{F}(\nabla u)^p w\,dx\right)^\frac{1}{p},\quad\forall\,u\in W_0^{1,p}(Ω,w)\leqno{\mathcal{(P)}} $$ where $W_0^{1,p}(Ω,w)$ is the weighted Sobolev space under a class of $p$-admissible weights $w$, where $f$ is some nonnegative integrable function in $Ω$. We discuss the case $0<q<1$ and observe that $$ μ(Ω):=\inf_{u\in W_{0}^{1,p}(Ω,w)}\Bigg\{\int_Ω\mathcal{F}(\nabla u)^p w\,dx:\int_Ω|u|^{q}f\,dx=1\Bigg\}\leqno{\mathcal{(Q)}} $$ is associated with singular weighted anisotropic $p$-Laplace equations. To this end, we also study existence and regularity properties of solutions for weighted anisotropic $p$-Laplace equations under the mixed and exponential singularities.

math.AP↗
arXiv2021

On an anisotropic p-Laplace equation with variable singular exponent

In this article, we study the following anisotropic p-Laplacian equation with variable exponent given by \begin{equation*} (P)\left\{\begin{split} -Δ_{H,p}u&=\frac{\la f(x)}{u^{q(x)}}+g(u)\text{ in }Ω,\\ u&>0\text{ in }Ω,\,u=0\text{ on }\partialΩ, \end{split}\right. \end{equation*} under the assumption $Ω$ is a bounded smooth domain in $\mathbb{R}^N$ with $p,N\geq 2$, $\la>0$ and $0<q \in C(\bar \Om)$. For the purely singular case that is $g\equiv 0$, we proved existence and uniqueness of solution. We also demonstrate the existence of multiple solution to $(P)$ provided $f\equiv 1$ and $g(u)=u^r$ for $r\in (p-1,p^*-1)$.

math.AP↗
arXiv2020

Bourgain-Brezis-Mironescu Domains

Bourgain et al.(2001) proved that for $p>1$ and smooth bounded domain $Ω\subseteq\mathbb{R}^N$, \begin{equation*} \lim\limits_{s\to1}(1-s)\iint \limits_{Ω\times Ω}\frac{\lvert f(x)-f(y) \rvert^p}{\lvert x-y \rvert^{N+sp}}dx dy=κ\int \limits_Ω\lvert \nabla f(x) \rvert^p dx \end{equation*} for all $f\in L^p(Ω)$. This gives a characterization of $W^{1,p}(Ω)$ by means of $W^{s,p}(Ω)$ seminorms only. For the case $p=1$, Dávila(2002) proved that when $Ω$ is a bounded domain with Lipschitz boundary, \begin{equation*} \lim\limits_{s\to1}(1-s)\iint \limits_{Ω\times Ω}\frac{\lvert f(x)-f(y) \rvert}{\lvert x-y \rvert^{N+s}}dx dy=κ[f]_{BV(Ω)} \end{equation*} for all $f\in L^1(Ω)$. This characterizes $BV(Ω)$ in terms of $W^{s,1}(Ω)$ seminorm. In this paper we extend the first result and partially extend the second result to extension domains.

math.AP↗