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Agnid Banerjee

Publications and source records attributed to Agnid Banerjee.

At least 37 records · Page 2Linked to original sources
arXiv2022

Quantitative uniqueness for fractional heat type operators

In this paper we obtain quantitative bounds on the maximal order of vanishing for solutions to $(\partial_t - Δ)^s u =Vu$ for $s\in [1/2, 1)$ via new Carleman estimates. Our main result Theorem 1.1 and Theorem 1.3 can be thought of as a parabolic generalization of the corresponding quantitative uniqueness result in the time independent case due to Rüland and it can also be regarded as a nonlocal generalization of a similar result due to Zhu for solutions to local parabolic equations.

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arXiv2022

Space-like strong unique continuation for some fractional parabolic equations

In this paper we establish the \emph{space-like} strong unique continuation for nonlocal equations of the type $(\partial_t - Δ)^s u= Vu$, for $0<s <1$. The proof of our main result, Theorem 1.1, is achieved via a conditional elliptic type doubling property for solutions to the appropriate extension problem, followed by a blowup analysis.

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arXiv2021

Lower semicontinuity and pointwise behavior of supersolutions for some doubly nonlinear nonlocal parabolic $p$-Laplace equations

We discuss pointwise behavior of weak supersolutions for a class of doubly nonlinear parabolic fractional $p$-Laplace equations which includes the fractional parabolic $p$-Laplace equation and the fractional porous medium equation. More precisely, we show that weak supersolutions have lower semicontinuous representative. We also prove that the semicontinuous representative at an instant of time is determined by the values at previous times. This gives a pointwise interpretation for a weak supersolution at every point. The corresponding results hold true also for weak subsolutions. Our results extend some recent results in the local parabolic case and in the nonlocal elliptic case to the nonlocal parabolic case. We prove the required energy estimates and measure theoretic De Giorgi type lemmas in the fractional setting.

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arXiv2021

Unified approach to $C^{1,α}$ regularity for quasilinear parabolic equations

In this paper, we are interested in obtaining a unified approach for $C^{1,α}$ estimates for weak solutions of quasilinear parabolic equations, the prototype example being \[ u_t - \text{div} (|\nabla u|^{p-2} \nabla u) = 0. \] without having to consider the singular and degenerate cases separately. This is achieved via a new scaling and a delicate adaptation of the covering argument developed by E.~DiBenedetto and A.~Friedman.

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arXiv2020

Carleman estimates for a class of variable coefficient degenerate elliptic operators with applications to unique continuation

In this paper, we obtain new Carleman estimates for a class of variable coefficient degenerate elliptic operators whose constant coefficient model at one point is the so called Baouendi-Grushin operator. This generalizes the results obtained by the two of us with Garofalo in [9] where similar estimates were established for the "constant coefficient" Baouendi-Grushin operator. Consequently, we obtain: (i) a Bourgain-Kenig type quantitative uniqueness result in the variable coefficient setting; (ii) and a strong unique continuation property for a class of degenerate sublinear equations. We also derive a subelliptic version of a scaling critical Carleman estimate proven by Regbaoui in the Euclidean setting using which we deduce a new unique continuation result in the case of scaling critical Hardy type potentials.

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arXiv2020

Some local properties of subsolutons and supersolutions for a doubly nonlinear nonlocal parabolic $p$-Laplace equation

We establish a local boundedness estimate for weak subsolutions to a doubly nonlinear parabolic fractional $p$-Laplace equation. Our argument relies on energy estimates and a parabolic nonlocal version of De Giorgi's method. Furthermore, by means of a new algebraic inequality, we show that positive weak supersolutions satisfy a reverse Hölder inequality. Finally, we also prove a logarithmic decay estimate for positive supersolutions.

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arXiv2020

Regularity for parabolic systems with critical growth in the gradient and applications

Motivated by problems arising in geometric flows, we prove several regularity results for systems of local and nonlocal equations, adapting to the parabolic case a neat argument due to Caffarelli. The geometric motivation of this work comes from recent works arising in the theory harmonic maps with free boundary in particular. We prove Hölder regularity of weak solutions.

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arXiv2020

Strong Backward uniqueness for sublinear parabolic equations

In this paper, we establish strong backward uniqueness for solutions to sublinear parabolic equations of the type (1.1). The proof of our main result Theorem 1.1 is achieved by means of a new Carleman estimate and a Weiss type monotonicity that are tailored for such parabolic sublinear operators.

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arXiv2020

On the finite $(Q-1)$-Hausdorff measure of the free boundary in the subelliptic obstacle problem

In this note, we prove the finite ${(Q-1)}$-Hausdorff measure of the free boundary in the obstacle problem in a Carnot group $\mathbb{G}$. Here, $Q$ represents the homogeneous dimension of $\mathbb{G}$. Our main result, Theorem 1.1, constitutes the subelliptic counterpart of the Euclidean result due to Caffarelli, but the analysis is complicated by the lack of commutation of the left-invariant vector fields. This obstruction is compensated by the use of right-invariant derivatives, and by a delicate compactness argument inspired to Caffarelli's fundamental works.

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arXiv2019

The Harnack inequality for a class of nonlocal parabolic equations

In this paper we establish a scale invariant Harnack inequality for the fractional powers of parabolic operators $(\partial_t - \mathscr{L})^s$, $0<s<1$, where $\mathscr{L}$ is the infinitesimal generator of a class of symmetric semigroups. As a by-product we also obtain a similar result for the nonlocal operators $(-\mathscr{L})^s$. Our focus is on non-Euclidean situations.

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arXiv2019

On a Bernoulli-type overdetermined free boundary problem

In this article we study a Bernoulli-type free boundary problem and generalize a work of Henrot and Shahgholian in \cite{HS1} to $\mathcal{A}$-harmonic PDEs. These are quasi-linear elliptic PDEs whose structure is modeled on the $p$-Laplace equation for a fixed $1 0$ is a given constant, then there exists a unique convex domain $Ω$ with $K\subset Ω$ and a function $u$ which is $\mathcal{A}$-harmonic in $Ω\setminus K$, has continuous boundary values $1$ on $\partial K$ and $0$ on $\partialΩ$, such that $|\nabla u|=c$ on $\partial Ω$. Moreover, $\partialΩ$ is $C^{1,γ}$ for some $γ>0$, and it is smooth provided $\mathcal{A}$ is smooth in $\mathbb{R}^n \setminus \{0\}$. We also show that the super level sets $\{u>t\}$ are convex for $t\in (0,1)$.

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arXiv2019

$C^{1, α}$ Regularity for degenerate fully nonlinear elliptic equations with Neumann boundary conditions

In this paper, we establish $C^{1, α}$ regularity upto the boundary for a class of degenerate fully nonlinear elliptic equations with Neumann boundary conditions. Our main result Theorem 2.1 constitutes the boundary analogue of the interior $C^{1, α}$ regularity result established in [21] for equations with similar structural assumptions. The proof of our main result is achieved via compactness arguments combined with new boundary Hölder estimates for equations which are uniformly elliptic when the gradient is either small or large.

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arXiv2019

The structure of the singular set in the thin obstacle problem for degenerate parabolic equations

We study the singular set in the thin obstacle problem for degenerate parabolic equations with weight $|y|^a$ for $a \in (-1,1)$. Such problem arises as the local extension of the obstacle problem for the fractional heat operator $(\partial_t - Δ_x)^s$ for $s \in (0,1)$. Our main result establishes the complete structure and regularity of the singular set of the free boundary. To achieve it, we prove Almgren-Poon, Weiss, and Monneau type monotonicity formulas which generalize those for the case of the heat equation ($a=0$).

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arXiv2019

The regular free boundary in the thin obstacle problem for degenerate parabolic equations

In this paper we study the existence, the optimal regularity of solutions, and the regularity of the free boundary near the so-called \emph{regular points} in a thin obstacle problem that arises as the local extension of the obstacle problem for the fractional heat operator $(\partial_t - Δ_x)^s$ for $s \in (0,1)$. Our regularity estimates are completely local in nature. This aspect is of crucial importance in our forthcoming work on the blowup analysis of the free boundary, including the study of the singular set. Our approach is based on first establishing the boundedness of the time-derivative of the solution. This allows reduction to an elliptic problem at every fixed time level. Using several results from the elliptic theory, including the epiperimetric inequality, we establish the optimal regularity of solutions as well as $H^{1+γ,\frac{1+γ}{2}}$ regularity of the free boundary near such regular points.

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