arXiv subjects

R. Lakshmi

Publications and source records attributed to R. Lakshmi.

6 recordsLinked to original sources

arXiv2026

Optimization for the first weighted eigenvalue of local-nonlocal operators with a potential

In this work, we study the optimization problem for the first eigenvalue of mixed local nonlocal operators plus a potential $V$. We begin by investigating the existence and fundamental properties of the eigenvalues, with special emphasis on the first eigenvalue. Finally, we discuss the dependence of the first eigenvalue on the potential function and establish the existence of optimal potentials within certain admissible classes. In particular, we show the existence of a unique maximizer and a minimizer of the first eigenvalue on any bounded, closed, and convex subset of $L^q(\Omega)$. Moreover, these results enable us to characterize the maximizers and minimizers in the closed unit ball of $L^q(\Omega)$, as well as in the class of rearrangements of any $V\in L^q(\Omega)$.

math.AP↗

arXiv2026

On weak and viscosity solutions to a nonhomogeneous mixed local-nonlocal equation

This paper explores the relationship between weak and viscosity solutions to a nonhomogeneous mixed local and non-local $p$-Laplace equation in a bounded Lipschitz domain in $\mathbb{R}^N$. Under certain conditions, we derive the comparison principle for weak subsolutions and weak supersolutions to the problem. For $1<p<\infty$, we establish that continuous weak supersolutions to the problem are viscosity supersolutions, using the comparison principle. Furthermore, we show that bounded viscosity supersolutions are weak supersolutions for $p \geq 2$.

math.AP↗

arXiv2026

Harnack inequality for superposition operators of mixed fractional order

The main aim of this paper is to establish the H\"older continuity and the Harnack inequality for weak solutions to Dirichlet problems associated with superposition operators of mixed fractional order, thereby complementing our previous work \cite{BGKL2026}. To achieve this, we extend the De Giorgi--Nash--Moser theory to the framework of superposition operators by introducing a novel {\it nonlocal superposition tail}, which appears to be the first contribution of its kind in the literature. The obtained results are new even in the classical linear case $p=2$, thereby illustrating the broader applicability of the analytical techniques developed in this work. As intermediate steps toward the proof of the main results, we also establish a logarithmic estimate for weak supersolutions, local boundedness for weak subsolutions, a weak Harnack inequality for weak supersolutions, an expansion of positivity for weak supersolutions, and tail estimates for weak solutions.

math.AP↗

arXiv2026

Regularity of superposition operators of mixed fractional order

We extend the De Giorgi--Nash--Moser theory to superposition operators of mixed fractional operators. In particular, we investigate several regularity properties for this class of operators. We establish the Caccioppoli-type inequality with tail for weak subsolutions, local boundedness of weak subsolutions, local H\"older continuity of weak solutions, the weak Harnack inequality for weak supersolutions, and the lower semicontinuity of weak supersolutions. Furthermore, we prove the expansion of positivity, a preliminary Harnack inequality, and the upper semicontinuity of weak subsolutions. Our results apply to both fixed-sign and sign-changing solutions involving mixed local--nonlocal superposition fractional operators. Notably, the results are new even in the classical linear case $p=2$, demonstrating the broader applicability of the techniques developed in this work.

math.AP↗

arXiv2025

On equivalence of weak and viscosity solutions to nonlocal double phase problems with nonhomogeneous data

This work focuses on the nonhomogeneous nonlocal double phase problem \begin{align*} L_au(x)=f(x,u,D_s^p u, D_{a,t}^q u) \text{ in } \Omega, \end{align*} where $\Omega\subset\mathbb{R}^N$ is a bounded domain with Lipschitz boundary, $0<s,t<1<p\leq q<\infty$ with $tq\leq sp$ and the operator $L_a$ is defined as \begin{align*} L_a u(x)&=2\operatorname{P.V.}\int_{\mathbb{R}^N}|u(x)-u(y)|^{p-2}(u(x)-u(y))K_{s,p}(x,y) &\ \ \ +2\operatorname{P.V.}\int_{\mathbb{R}^N}a(x,y)|u(x)-u(y)|^{q-2}(u(x)-u(y))K_{t,q}(x,y)dy. \end{align*} We establish the equivalence between weak and viscosity solutions under boundedness and continuity assumptions. In addition, the local boundedness of weak solutions in some special cases on $f$ is also obtained using the notion of De Giorgi classes.

math.AP↗

arXiv2024

A weighted eigenvalue problem for mixed local and nonlocal operators with potential

We study an {\it indefinite weighted eigenvalue problem} for an operator of {\it mixed-type} (that includes both the classical {\it $p$-Laplacian} and the {\it fractional $p$-Laplacian}) in a bounded open subset $\Omega\subset \mathbb{R}^N \,(N\geq2)$ with {\it Lipschitz boundary} $\partial \Omega$, which is given by \begin{align*} -\Delta_p u + (-\Delta_p)^su+V(x)|u|^{p-2}u&=\lambda g(x)|u|^{p-2}u~\text{in}~\Omega, u&=0~\text{in}~\mathbb{R}^N\setminus\Omega, \end{align*} where $\lambda >0$ is a parameter, exponents $0 0$ a.e. in $\Omega$. Using the variational tools together with a {\it weak comparison} and {\it strong maximum principles}, we investigate the existence and uniqueness of {\it principal eigenvalue} and discuss its qualitative properties. Moreover, with the help of {\it Ljusternik-Schnirelman category theory}, it is proved that there exists a {\it nondecreasing sequence of positive eigenvalues} which goes to infinity. Further, we show that {\it the set of all positive eigenvalues is closed}, and {\it eigenfunctions} associated with every {\it positive eigenvalue} are bounded.

math.AP↗