Agnid Banerjee

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Compactness methods for $$\Gamma ^{1,\alpha }$$ Γ1,α boundary Schauder estimates in Carnot groups

Agnid Banerjee, Nicola Garofalo, Isidro H. Munive

Calculus of Variations and Partial Differential Equations > 2019 > 58 > 3 > 1-29

The aim of this paper is to prove $$\Gamma ^{1,\alpha }$$ Γ1,α Schauder estimates near a $$C^{1,\alpha }$$ C1,α non-characteristic portion of the boundary for $$\Gamma ^{0, \alpha }$$ Γ0,α perturbations of horizontal Laplaceans in Carnot groups. This situation of minimally smooth domains presents itself naturally in the study of subelliptic free boundary problems of obstacle type, see Danielli...

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Monotonicity of generalized frequencies and the strong unique continuation property for fractional parabolic equations

Agnid Banerjee, Nicola Garofalo

Advances in Mathematics > 2018 > 336 > C > 149-241

We study the strong unique continuation property backwards in time for the nonlocal equation in Rn+1(0.1)(∂t−Δ)su=V(x,t)u,s∈(0,1). Our main result Theorem 1.2 can be thought of as the nonlocal counterpart of the result obtained in [30] for the case when s=1. In order to prove Theorem 1.2 we develop the regularity theory of the extension problem for the equation (0.1). With such theory in hands we...

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Higher regularity of the free boundary in the parabolic Signorini problem

Agnid Banerjee, Mariana Smit Vega Garcia, Andrew K. Zeller

Calculus of Variations and Partial Differential Equations > 2017 > 56 > 1 > 1-26

We show that the quotient of two caloric functions which vanish on a portion of an $$H^{k+ \alpha }$$ H k + α regular slit is $$H^{k+ \alpha }$$ H k + α at the slit, for $$k \ge 2$$ k ≥ 2 . In the case $$k=1$$ k = 1 , we show that the quotient is in $$H^{1+\alpha }$$ H 1 + α if the slit is assumed to be space-time $$C^{1, \alpha }$$ ...

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Quantitative uniqueness for elliptic equations at the boundary of C1,Dini domains

Agnid Banerjee, Nicola Garofalo

Journal of Differential Equations > 2016 > 261 > 12 > 6718-6757

Based on a variant of the frequency function approach of Almgren, we establish an optimal upper bound on the vanishing order of solutions to variable coefficient Schrödinger equations at a portion of the boundary of a C1,Dini domain. Such bound provides a quantitative form of strong unique continuation at the boundary. It can be thought of as a boundary analogue of an interior result recently obtained...

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A parabolic analogue of the higher-order comparison theorem of De Silva and Savin

Agnid Banerjee, Nicola Garofalo

Journal of Differential Equations > 2016 > 260 > 2 > 1801-1829

We show that the quotient of two caloric functions which vanish on a portion of the lateral boundary of a Hk+α domain is Hk+α up to the boundary for k≥2. In the case k=1, we show that the quotient is in H1+α if the domain is assumed to be space-time C1,α regular. This can be thought of as a parabolic analogue of a recent important result in [8], and we closely follow the ideas in that paper. We also...

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Modica type gradient estimates for an inhomogeneous variant of the normalized p-Laplacian evolution

Agnid Banerjee, Nicola Garofalo

Nonlinear Analysis: Theory, Methods & Applications > 2015 > 121 > C > 458-468

In this paper, we study an inhomogeneous variant of the normalized p-Laplacian evolution which has been recently treated in Banerjee and Garofalo (2013), Does (2011), Manfredi et al. (2010), and Juutinen (2014). We show that if the initial datum satisfies the pointwise gradient estimate (1.6) a.e., then the unique solution to the Cauchy problem (1.2) satisfies the same gradient estimate a.e. for all...

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Boundary behavior of nonnegative solutions of fully nonlinear parabolic equations

Agnid Banerjee, Nicola Garofalo

manuscripta mathematica > 2015 > 146 > 1-2 > 201-222

In this paper we consider fully nonlinear parabolic equations in $${\mathbb{R}^{n+1}}$$ R n + 1 of the type $$F(D^2 u, Du, x, t) = u_{t},\quad \quad \quad {(0.1)}$$ F ( D 2 u , D u , x , t ) = u t , ( 0.1 ) where F satisfies the structural conditions in (2.2) below, along with F(0, 0, x, t) = 0. Viscosity solutions to (0.1) are a subclass of the (viscosity) solutions...

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Gradient bounds for p-harmonic systems with vanishing Neumann (Dirichlet) data in a convex domain

Agnid Banerjee, John L. Lewis

Nonlinear Analysis > 2014 > 100 > Complete > 78-85

Let Ω̃ be a bounded convex domain in Euclidean n space, xˆ∈∂Ω̃, and r>0. Let ũ=(ũ1,ũ2,…,ũN) be a weak solution to ∇⋅(|∇ũ|p−2∇ũ)=0in Ω̃∩B(xˆ,4r)with |∇ũ|p−2ũν=0on ∂Ω̃∩B(xˆ,4r). We show that sub solution type arguments for certain uniformly elliptic systems can be used to deduce that |∇ũ| is bounded in Ω̃∩B(xˆ,r) with constants depending only on n,p,N, and rn|Ω̃∩B(xˆ,r)|. Our argument...

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A remark on a vanishing property for viscosity solutions of fully nonlinear parabolic equations

Agnid Banerjee

Nonlinear Analysis > 2013 > 80 > Complete > 14-17

We make the observation that, under some natural conditions on F (stated in (A)–(C) in the main text), if a viscosity solution of the fully nonlinear parabolic equation F(D2u,Du,u)−ut=0 vanishes to infinite order at (x0,t0), then there is a small spatial neighborhood Br0(x0)×{t0} of (x0,t0) in which u vanishes identically. The proof is inspired in an essential way by the ideas employed in the recent...

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Search results for: Agnid Banerjee

Compactness methods for $$\Gamma ^{1,\alpha }$$ Γ1,α boundary Schauder estimates in Carnot groups

Monotonicity of generalized frequencies and the strong unique continuation property for fractional parabolic equations

Higher regularity of the free boundary in the parabolic Signorini problem

Quantitative uniqueness for elliptic equations at the boundary of C1,Dini domains

A parabolic analogue of the higher-order comparison theorem of De Silva and Savin

Modica type gradient estimates for an inhomogeneous variant of the normalized p-Laplacian evolution

Boundary behavior of nonnegative solutions of fully nonlinear parabolic equations

Gradient bounds for p-harmonic systems with vanishing Neumann (Dirichlet) data in a convex domain

A remark on a vanishing property for viscosity solutions of fully nonlinear parabolic equations

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