arXiv subjects

Stuti Das

Publications and source records attributed to Stuti Das.

7 recordsLinked to original sources

arXiv2026

Regularity for the fractional logarithmic $p$-Laplacian

We prove the Harnack inequality (with tails) and local H\"older regularity for the fractional logarithmic $p$-Laplace operator, which is derived by differentiating the fractional $p$-Laplace operator with respect to its order. To be more precise, for a suitable function $u,$ the operator reads as the first order derivative \begin{align*} (-\Delta_p)^{s+\log} u:= \frac{{\rm d}}{{\rm d}t}(-\Delta_p)^t u \Big|_{t=s} \end{align*} at any arbitrary order $s\in (0, 1).$ The kernel of this operator involves a logarithmic factor. As a consequence, it changes sign at large scales and, near the diagonal, is more singular than the kernel of the fractional $p$-Laplacian. To achieve our regularity estimates, we adopt the classical De Giorgi-Nash-Moser techniques in this setting. We also construct an example showing that the Harnack inequality fails without tail terms. Our results are new even in the linear setup $p=2$.

math.AP↗

arXiv2026

Harnack inequality for mixed local-nonlocal weighted homogeneous equations

We consider the following class of mixed local-nonlocal equations: \begin{equation}\tag{P}\label{eq:P} -\Delta_p u + (-\Delta)_p^s u = V |u|^{p-2}u \text{ in } \Omega, \end{equation} where $s \in (0,1), p \in (1, \infty)$, and the weight function $V$ lies in scaling subcritical Lebesgue space $L^q(\Omega)$ where $q>d/p$ when $d>p$ and $q>1$ when $d \le p$. We establish the Harnack inequality for a weak solution and the weak Harnack inequality for a weak supersolution to \eqref{eq:P}. Our approach is based on the De Giorgi-Nash-Moser theory, the expansion of positivity and estimates involving a tail term. Our results also apply to integro-differential operators, with the prototype given by $(-\Delta)_p^s$. This work generalizes some regularity results of Garain-Kinnunen (Trans. Am. Math. Soc., 375(8), 2022) and Garain (Nonlinear Anal., 256, 2025) to the setting of general weight functions.

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arXiv2025

Regularity and existence for a mixed local-nonlocal parabolic equation with variable singularities and measure data

This article proves the existence, non-existence, regularity and asymptotic behavior of weak solutions for a class of mixed local-nonlocal parabolic problems involving singular nonlinearities and measure data extending the works of \cite{sanjitgarain,lazermc}. A central contribution of this work is the inclusion of a variable singular exponent. We examine both the purely singular and perturbed singular cases in the context of measure-valued data, where the source terms can simultaneously take the form of measures. To the best of our knowledge, this phenomenon is new, even in the case of a constant singular exponent involving only a local operator. Further, all our results are also true for the operator being local only.

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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.

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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} -\Delta_pu+(-\Delta)_p^s u&=\frac{\lambda}{u^{\gamma}}+u^r \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 )_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 $-\Delta_p$ is the usual $p$-Laplace operator. Under the assumptions that $\Omega$ is a bounded domain in $\mathbb{R}^{n}$ with regular enough boundary, $p>1$, $n> p$, $s\in(0,1)$, $\lambda>0$ 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 $0<\gamma<1$ and some certain values of $\lambda$. Further, for every $\gamma>0$, assuming strict convexity of $\Omega$, 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 $\lambda$, extending the result of \cite{garaingeometric}. Here $c_{n,s}$ is a suitable normalization constant, and $\operatorname{P.V.}$ stands for Cauchy Principal Value.

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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-\Delta u+(-\Delta)^s u&=\frac{f(x,t)}{u^{\gamma(x,t)}} \text { in } \Omega_T:=\Omega \times(0, T), \\ u&=0 \text { in }(\mathbb{R}^n \backslash \Omega) \times(0, T), \\ u(x, 0)&=u_0(x) \text { in } \Omega ; \end{split} \end{eqnarray} where \begin{equation*} (-\Delta )^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 $\gamma$ is a positive continuous function on $\overline{\Omega}_T$ and $\Omega$ 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.

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arXiv2023

Gradient H\"older regularity in mixed local and nonlocal linear parabolic problem

We prove the local H\"older regularity of weak solutions to the mixed local nonlocal parabolic equation of the form \begin{equation*} u_t-\Delta u+\text{P.V.}\int_{\mathbb{R}^{n}} {\frac{u(x,t)-u(y,t)}{{\left|x-y\right|}^{n+2s}}}dy=0, \end{equation*} where $0<s<1$; for every exponent $\alpha_0\in(0,1)$. Here, $\Delta$ is the usual Laplace operator. Next, we show that the gradients of weak solutions are also $\alpha$-H\"older continuous for some $\alpha\in (0,1)$. Our approach is purely analytic and it is based on perturbation techniques.

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