arXiv subjects
Publications and source records attributed to Debraj Kar.
We prove Harnack inequality with an optimal tail for linear mixed local-nonlocal parabolic equation on the Heisenberg group. For this purpose, we make use of the expansion of positivity method and a clustering lemma, thereby bypassing Moser iteration and logarithmic estimates.
We prove the H\"older continuity for weak solutions to parabolic p-Laplace equations on the Heisenberg group. We deduce this result while considering an optimal tail condition.
We prove local boundedness for a quasilinear parabolic equation on the Heisenberg group \[ \partial_t u(\xi,t) + \text{p.v.}\int_{\mathbb{H}^N} \frac{|u(\xi,t)-u(\eta,t)|^{p-2}(u(\xi,t)-u(\eta,t))}{|\eta^{-1}\circ \xi|^{Q+sp}} \,d\eta = 0, \] with optimal regularity assumption on the tail term. We also prove interpolation inequalities and an extension theorem for fractional Sobolev spaces on the Heisenberg group.
In this paper we prove local gradient estimates and higher differentiability result for the solutions of variational obstacle inequalities \int_\Omega\big<\mathcal{A}(x,u,Du),D(\phi-u)\big>dx\geq \int_\Omega\mathcal{B}(x,u,Du)(\phi-u)dx. for all $\phi\in \mathcal{K}_\psi(\Omega)$. Here $\Omega(\subset\mathbb{R}^n)$ is bounded, $n\geq 2$ and $\psi:\Omega\rightarrow\mathbb{R}$ is called obstacle. Here we deal with variable exponent growth , namely $p(.)$-growth . At first we prove Calder\'on-Zygmund estiamte and then using this result to prove higher differentiability result in Besov scale.