arXiv subjects

Souvik Bhowmick

Publications and source records attributed to Souvik Bhowmick.

7 recordsLinked to original sources

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

Superlinear problems involving nonlinear superposition operators of mixed fractional order

In this work, we study a class of elliptic problems involving nonlinear superpositions of fractional operators of the form \[ A_{\mu,p}u := \int_{[0,1]} (-\Delta)_{p}^{s} u \, d\mu(s), \] where $\mu$ is a signed measure on $[0,1]$, coupled with nonlinearities of superlinear type. Our analysis covers a variety of superlinear growth assumptions, beginning with the classical Ambrosetti--Rabinowitz condition. Within this framework, we construct a suitable variational setting and apply the Fountain Theorem to establish the existence of infinitely many weak solutions. The results obtained are novel even in the special cases of superpositions of fractional $p$-Laplacians, or combinations of the fractional $p$-Laplacian with the $p$-Laplacian. More generally, our approach applies to finite sums of fractional $p$-Laplacians with different orders, as well as to operators in which fractional Laplacians appear with ``wrong'' signs. A distinctive contribution of the paper lies in providing a unified variational framework that systematically accommodates this broad class of operators.

math.AP↗

arXiv2025

Infinitely many solutions for nonlinear superposition operators of mixed fractional order involving critical exponent

This paper addresses a class of elliptic problems involving the superposition of nonlinear fractional operators with the critical Sobolev exponent in the sublinear regimes. We establish the existence of infinitely many nontrivial weak solutions using a variational framework combining a truncation argument with the notion of genus. A central part of our analysis is the verification of the Palais--Smale (PS) condition for the associated energy functional for every $q \in (1, p_{s_\sharp}^*)$, despite the challenges posed by the lack of compactness due to the critical exponent. The results obtained in the paper are new even in the classical case $p = 2$, highlighting the broader applicability of the methods developed here.

math.AP↗

arXiv2025

On sign-changing solutions for mixed local and nonlocal $p$-Laplace operator

In this paper, we use the method of invariant sets of descending flows to demonstrate the existence of multiple sign-changing solutions for a class of elliptic problems with zero Dirichlet boundary conditions. By combining Nehari manifold techniques with a constrained variational approach and Brouwer degree theory, we establish the existence of a least-energy sign-changing solution. Furthermore, we prove that the energy of the least energy sign-changing solution is strictly greater than twice the ground state energy. This work extends the celebrated results of Bartsch $et~al.$ [Proc. Lond. Math. Soc. (3), 91(1): 129-152, 2005] and Chang $et~al.$ [Adv. Nonlinear Stud., 19(1): 29-53, 2019] to the mixed local and nonlocal $p$-Laplace operator, providing a novel contribution even in the case when $p=2$.

math.AP↗

arXiv2024

Existence results for mixed local and nonlocal elliptic equations involving singularity and nonregular data

In this paper, we prove the existence of weak, veryweak and duality solutions to a class of elliptic problems involving singularity and measure data which is given by: $-\Delta u+(-\Delta)^s u = \frac{f(x)}{u^\gamma} +\mu$ in $\Omega$ with the zero Dirichlet boundary data $u=0$ in $\mathbb R^N \setminus \Omega$. The existence of weak solutions is obtained by approximating a sequence of problems for $0<\gamma\leq1$ and $\gamma>1$. We employ Schauder's fixed point theorem and embeddings of Marcinkiewicz spaces. The novelty of our work is that we prove the existence of a duality solution and its equivalence with weak solutions to the problem $\mathcal{L}u=\mu$. Moreover, we prove a veryweak maximum principle and a Kato-type inequality for the mixed local-nonlocal operator $\mathcal{L}=-\Delta +(-\Delta)^s$, which are crucial tools to guarantee the existence of veryweak solutions to the problem. Using a Kato-type inequality, maximum principle together with sub-super solution method, we prove the existence of veryweak solution for $0<\gamma<1$. Our work extends the studies due to Oliva and Petitta [ESAIM Control Optim. Calc. Var., 22(1):289--308, 2016.] and Petitta [Adv. Nonlinear Stud., 16(1):115--124, 2016.] for the mixed local-nonlocal operator.

math.AP↗

arXiv2014

Bangla Text Recognition from Video Sequence: A New Focus

Extraction and recognition of Bangla text from video frame images is challenging due to complex color background, low-resolution etc. In this paper, we propose an algorithm for extraction and recognition of Bangla text form such video frames with complex background. Here, a two-step approach has been proposed. First, the text line is segmented into words using information based on line contours. First order gradient value of the text blocks are used to find the word gap. Next, a local binarization technique is applied on each word and text line is reconstructed using those words. Secondly, this binarized text block is sent to OCR for recognition purpose.

cs.CV↗