Agnid Banerjee

• PhD
• Associate Professor at Arizona State University

We prove a theorem of unique continuation in measure for nonlocal equations of the type $(\partial_t - \Delta)^s u= V(x,t) u$, for $0<s <1$. Our main result, Theorem 1.1, establishes a delicate nonlocal counterpart of the unique continuation in measure for the local case $s=1$.

In his seminal 1943 paper F. Rellich proved that, in the complement of a cavity \(\Omega = \{x\in \mathbb {R}^n\mid |x|>R_0\}\), there exist no nontrivial solution f of the Helmholtz equation \(\Delta f = - \lambda f\), when \(\lambda >0\), such that \(\int _{\Omega } |f|^2 dx < \infty \). In this note we generalise this result by showing that if \...

In this paper, we introduce and analyse an explicit formulation of fractional powers of the parabolic Lamé operator and we then study the extension problem associated to such non-local operators. We also study the various regularity properties of solutions to such an extension problem via a transformation as in Ang et al. (Commun Partial Differ Equ...

In his seminal 1981 study D. Jerison showed the remarkable negative phenomenon that there exist, in general, no Schauder estimates near the characteristic boundary in the Heisenberg group Hn.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \u...

We establish a new sharp estimate of the order of vanishing of solutions to parabolic equations with variable coefficients. For real-analytic leading coefficients, we prove a localised estimate of the nodal set, at a given time-level, that generalises the celebrated one of Donnelly and Fefferman. We also establish Landis type results for global sol...

For $s\in [\tfrac {1}{2},\, 1)$, let $u$ solve $(\partial _t - \Delta )^s u = Vu$ in $\mathbb {R}^{n} \times [-T,\, 0]$ for some $T>0$ where $||V||_{ C^2(\mathbb {R}^n \times [-T, 0])} < \infty$. We show that if for some $0<\mathfrak {K} < T$ and $\epsilon >0$\[ {\unicode{x2A0D}}-_{[-\mathfrak{K},\, 0]} u^2(x, t) {\rm d}t \leq Ce^{-|x|^{2+\epsilon}...

For $s \in [1/2, 1)$, let $u$ solve $(\partial_t - \Delta)^s u = Vu$ in $\mathbb R^{n} \times [-T, 0]$ for some $T>0$ where $||V||_{ C^2(\mathbb R^n \times [-T, 0])} < \infty$. We show that if for some $0< c< T$ and $\epsilon>0$ $$\frac{1}{c} \int_{[-c,0]} u^2(x, t) dt \leq Ce^{-|x|^{2+\epsilon}}\ \forall x \in \mathbb R^n,$$ then $u \equiv 0$ in $...

In this short note, we prove that if u solves \((\partial _t - \Delta )^s u = Vu\) in \({\mathbb {R}}^n_x \times {\mathbb {R}}_t\), and vanishes to infinite order at a point \((x_0, t_0)\), then \(u \equiv 0\) in \({\mathbb {R}}^n_x \times {\mathbb {R}}_t\). This sharpens (and completes) our earlier result that proves \(u(\cdot , t) \equiv 0\) for...

In this paper we obtain quantitative bounds on the maximal order of vanishing for solutions to (∂t-Δ)su=Vu\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\partial _t -...

In this paper, we prove gradient continuity estimates for viscosity solutions to $$\Delta _{p}^N u - u_t= f$$ Δ p N u - u t = f in terms of the scaling critical $$L(n+2,1 )$$ L ( n + 2 , 1 ) norm of f , where $$\Delta _{p}^N$$ Δ p N is the game theoretic normalized $$p-$$ p - Laplacian operator defined in (1.2) below. Our main result, Theorem 2.5 c...

We show that if $u$ solves the fractional parabolic equation $(\partial_t - \Delta)^s u = Vu$ in $Q_5:=B_5 \times (-25, 0]$ ($0<s<1$) such that $u(\cdot, 0) \not\equiv 0$, then the maximal vanishing order of $u$ in space-time at $(0,0)$ is upper bounded by $C\left(1+\|V\|_{C^{1}_{(x,t)}}^{1/2s}\right)$. As $s \to 1$, it converges to the sharp maxim...

In this paper, we are interested in obtaining a unified approach to C^{1,\alpha} estimates for weak solutions of quasilinear parabolic equations, the prototype example being u_t - \operatorname{div} \big(|\nabla u|^{p-2} \nabla u\big) = 0. without having to consider the singular and degenerate cases separately. This is achieved via a new scaling an...

In this short note we prove that if $u$ solves $(\partial_t - \Delta)^s u = Vu$ in $\mathbb R^n_x \times \mathbb R_t$, and vanishes to infinite order at a point $(x_0, t_0)$, then $u \equiv 0$ in $\mathbb R^n_x \times \mathbb R_t$. This sharpens (and completes) our earlier result that proves $u(\cdot, t) \equiv 0$ for $t \leq t_0$ if it vanishes to...

In this paper, we prove gradient continuity estimates for viscosity solutions to $\Delta_{p}^N u- u_t= f$ in terms of the scaling critical $L(n+2,1 )$ norm of $f$, where $\Delta_{p}^N$ is the game theoretic normalized $p-$Laplacian operator defined in (1.2) below. Our main result, Theorem 2.5 constitutes borderline gradient continuity estimate for...

In his seminal 1981 study D. Jerison showed the remarkable negative phenomenon that there exist, in general, no Schauder estimates near the characteristic boundary in the Heisenberg group $\mathbb H^n$. On the positive side, by adapting tools from Fourier and microlocal analysis, he developed a Schauder theory at a non-characteristic portion of the...

In this paper, we establish the optimal interior regularity and the C1,γ smoothness of the regular part of the free boundary in the thin obstacle problem for a class of degenerate elliptic equations with variable coefficients.

In this paper we establish the space-like strong unique continuation for nonlocal equations of the type (∂t−Δ)su=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.

In this paper, we introduce and analyse an explicit formulation of fractional powers of the parabolic Lam\'e operator $\mathbb{H}$ and we then study the extension problem associated to such non-local operators. We also study the various regularity properties of solutions to such an extension problem via a transformation which reduces the extension...

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 regul...

We obtain sharp maximal vanishing order at a given time level for solutions to parabolic equations with a $C{^1}$ potential $V$. Our main result Theorem 1.1 is a parabolic generalization of a well known result of Donnelly-Fefferman and Bakri. It also sharpens a previous result of Zhu that establishes similar vanishing order estimates which are inst...

In this paper, we prove the strong unique continuation property at the origin for solutions of the following scaling critical parabolic differential inequality \[ |\operatorname{div} (A(x,t) \nabla u) - u_t| \leq \frac{M}{|x|^{2}} |u|,\ \ \ \ \] where the coefficient matrix $A$ is Lipschitz continuous in $x$ and $t$. Our main result sharpens a prev...

We study an inverse problem for variable coefficient fractional parabolic operators of the form $(\partial_t -\operatorname{div}(A(x) \nabla_x)^s + q(x,t)$ for $s\in(0,1)$ and show the unique recovery of $q$ from exterior measured data. Similar to the fractional elliptic case, we use Runge type approximation argument which is obtained via a global...

We discuss pointwise behavior of weak supersolutions for a class of doubly nonlinear parabolic fractional [Formula: see text]-Laplace equations which includes the fractional parabolic [Formula: see text]-Laplace equation and the fractional porous medium equation. More precisely, we show that weak supersolutions have lower semicontinuous representat...

In this note we give an elementary proof of the space-like real analyticity of solutions to a degenerate evolution problem that arises in the study of fractional parabolic operators of the type $(\partial_t - div_x(B(x)\nabla_x))^s$, $0<s<1$. Our primary interest is in the so-called \emph{extension variable}. We show that weak solutions that are ev...

In this paper we obtain quantitative bounds on the maximal order of vanishing for solutions to $(\partial_t - \Delta)^s u =Vu$ for $s\in [1/2, 1)$ via new Carleman estimates. Our main result Theorem 1.1 can be thought of as a parabolic generalization of the corresponding quantitative uniqueness result in the time independent case due to R\"uland an...

In this paper we establish the \emph{space-like} strong unique continuation for nonlocal equations of the type $(\partial_t - \Delta)^s u= Vu$, 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.

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 inequal...

In this article we study a Bernoulli-type free boundary problem and generalize a work of Henrot and Shahgholian in [25] to \(\mathcal{A}\)-harmonic PDEs. These are quasi-linear elliptic PDEs whose structure is modelled on the \(p\)-Laplace equation for a fixed \(1<p<\infty\). In particular, we show that if \(K\) is a bounded convex set satisfying t...

We study the singular set in the thin obstacle problem for degenerate parabolic equations with weight |y|a\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|y|^a$$\end{do...

This paper is devoted to the existence, the optimal regularity of solutions, and the regularity of the free boundary near the so-called regular points in a thin obstacle problem that arises as the local extension of the obstacle problem for the fractional heat operator ( ∂ t − Δ x ) s (\partial _t - \Delta _x)^s for s ∈ ( 0 , 1 ) s \in (0,1) . The...

In this paper, we establish C1,α regularity up to 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 C1,α regularity result established in Imbert and Silvestre (Adv. Math. 233: 196–206, 2013) for equations with simi...

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 semi...

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 [10] where similar estimates were established for the "constant coefficien...

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 1.3 is achieved by means of a new Carleman estimate and a Weiss type monotonicity formula that are tailored for such parabolic sublinear operators.

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...

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 H\"older ine...

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 − div(|∇u| p−2 ∇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 argu...

In this paper, we prove borderline gradient continuity of viscosity solutions to fully nonlinear elliptic equations at the boundary of a C^{1,\mathrm{Dini}} -domain. Our main result constitutes the boundary analogue of the borderline interior gradient regularity estimates established by P. Daskalopoulos, T. Kuusi and G. Mingione. We however mention...

In this paper we establish the optimal interior regularity and the $C^{1,\gamma}$ smoothness of the regular part of the free boundary in the thin obstacle problem for a class of degenerate elliptic equations with variable coefficients.

In this paper, we prove H2+α regularity for viscosity solutions to non-convex fully nonlinear parabolic equations near the boundary. This constitutes the parabolic counterpart of a similar C2,α regularity result due to Silvestre and Sirakov proved in [17] for solutions to non-convex fully nonlinear elliptic equations.

In this note we prove the strong unique continuation property at the origin for the solutions of the parabolic differential inequality |Δu-ut|≤M|x|2|u|,with the critical inverse square potential. Our main result sharpens a previous one of Vessella concerned with the subcritical case.

In this paper, we establish a scale invariant Harnack inequality for the fractional powers of parabolic operators [Formula: see text], [Formula: see text], where [Formula: see text] is the infinitesimal generator of a class of symmetric semigroups. As a by-product, we also obtain a similar result for the nonlocal operators [Formula: see text]. Our...

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 H\"older reg...

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.

In this note we prove the strong unique continuation property at the origin for the solutions of the parabolic differential inequality \[ |\Delta u - u_t| \leq \frac{M}{|x|^2} |u|, \] with the critical inverse square potential. Our main result sharpens a previous one of Vessella concerned with the subcritical case.

In this paper, we establish some new L2−L2 Carleman estimates for the Baouendi–Grushin operators Bγ, in Equation (1). We apply such estimates to obtain: (i) an extension of the Bourgain–Kenig quantitative unique continuation and (ii) the strong unique continuation property for some degenerate sublinear 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 o...

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 $1<p<\infty$. In particular, we show that if $K$ is a bounded convex set satisfying the...

In this paper, we obtain gradient continuity estimates for viscosity solutions of [Formula: see text] in terms of the scaling critical [Formula: see text]-norm of [Formula: see text], where [Formula: see text] is the normalized [Formula: see text]-Laplacian operator. Our main result corresponds to the borderline gradient continuity estimate in term...

In this paper, we prove $\mathcal{H}^{2+\alpha}$ regularity for viscosity solutions to non-convex fully nonlinear parabolic equations near the boundary. This constitutes the parabolic counterpart of a similar $C^{2, \alpha}$ regularity result due to Silvestre and Sirakov proved in [15] for solutions to non-convex fully nonlinear elliptic 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 $(\partial_t - \Delta_x)^s$ for $s \in (0,1)$. Our regularity estimates are...

In this paper, we prove the strong unique continuation property for the following degenerate elliptic equation $$\begin{aligned} \Delta _zu +|z|^2\partial _t^2u = Vu,\quad (z,t) \in {\mathbb {R}}^N \times {\mathbb {R}} \end{aligned}$$ (0.1) where the potential V satisfies either of the following growth assumptions $$\begin{aligned}&\left| V(z,t) \r...

In this paper, we obtain gradient continuity estimates for viscosity solutions of $\Delta_{p}^N u= f$ in terms of the scaling critical $L(n,1 )$ norm of $f$, where $\Delta_{p}^N$ is the normalized $p-$Laplacian operator defined in (1.2) below. Our main result, Theorem 2.2, corresponds to the borderline gradient continuity estimate in terms of the m...

In this paper, we establish $C^{1, \alpha}$ 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 $C^{1, \alpha}$ regularity result established in [18] for equations with similar structural assumptions....

In this paper, by using elementary arguments based on integration by parts, we establish a new $L^{2}-L^{2}$ type Carleman estimate with singular weights for Baouendi-Grushin operators $\mathbb{B}_{\gamma}$ which implies strong unique continuation for stationary Schr\"odinger equations associated to $\mathbb{B}_{\gamma}$. Subsequently by establishi...

We study the singular set in the thin obstacle problem (1.1) 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 $(\partial_t - \Delta)^s$ for $s \in (0,1)$, see the recent work [2]. Our main result establishes the complete str...

In this paper, we establish space like strong unique continuation property (sucp) for uniformly parabolic sublinear equations under appropriate structural assumptions. Our main result Theorem 1.1 constitutes the parabolic counterpart of the strong unique continuation result recently established by Ruland in [Ru] for analogous elliptic sublinear equ...

In this paper, we establish space like strong unique continuation property for uniformly parabolic sublinear equations under appropriate structural assumptions. Our main result Theorem 1.1 constitutes the parabolic counterpart of the strong unique continuation result recently established in (Ruland, J. Differential Equations 265 (2018) 6009–6035) f...

In this paper we prove strong unique continuation for the following stationary Schr\"odinger equation \begin{equation}\label{e0} \Delta_zu +|z|^2\partial_t^2u = Vu,\quad (z,t) \in \mathbb{R}^N \times \mathbb{R} \end{equation} where the potential $V$ satisfies either of the following growth assumptions \begin{align} & |V(z,t)| \leq \frac{f(\rho(z,t)...

In this paper, we prove borderline gradient continuity of viscosity solutions to Fully nonlinear elliptic equations at the boundary of a $C^{1,\dini}$-domain. Our main result Theorem 3.1 is a sharpening of the boundary gradient estimate proved by Ma-Wang following the borderline interior gradient regularity estimates established Daskalopoulos-Kuusi...

The aim of this paper is to prove $\Gamma^{1,\alpha}$ Schauder estimates near a $C^{1,\alpha}$ non-characteristic portion of the boundary for perturbations of horizontal Laplaceans in Carnot groups. By covering the intermediate situation of less regular domains and equations with less regular coefficients, our main result, Theorem 1.1, complements...

The aim of this paper is to prove $\Gamma^{1,\alpha}$ Schauder estimates near a $C^{1,\alpha}$ non-characteristic portion of the boundary for $\Gamma^{0, \alpha}$ 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 obstacl...

We extend the symmetry result of Serrin and Weinberger from the Laplacian operator to the highly degenerate game-theoretic $p$-Laplacian operator and show that viscosity solutions of $-\Delta_p^Nu=1$ in $\Omega$, $u=0$ and $\tfrac{\partial u}{\partial\nu}=-c\neq 0$ on $\partial\Omega$ can only exist on a bounded domain $\Omega$ if $\Omega$ is a bal...

We study the strong unique continuation property backwards in time for the nonlocal equation in Rⁿ⁺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...

Based on a variant of the frequency function approach of Almgren([Al]), we establish an optimal upper bound on the vanishing order of solutions to stationary Schr\"odinger equations associated to sub-Laplacian on a Carnot group of arbitrary step $\mathbb{G}$. Such bound provides a quantitative form of strong unique continuation and can be thought o...

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

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 Schr\"odinger equations at a portion of the boundary of a $C^{1,Dini}$ domain. Such bound provides a quantitative form of strong unique continuation at the boundary. It can be thought of a...

Based on a variant of the frequency function approach of Almgren, we establish an optimal bound on the vanishing order of solutions to stationary Schr\"odinger equations associated to a class of subelliptic equations with variable coefficients whose model is the so-called Baouendi-Grushin operator. Such bound provides a quantitative form of strong...

We show that the quotient of two caloric functions which vanish on a portion of the lateral boundary of a $H^{k+ \alpha}$ domain is $H^{k+ \alpha}$ up to the boundary for $k \geq 2$. In the case $k=1$, we show that the quotient is in $H^{1+\alpha}$ if the domain is assumed to be space-time $C^{1, \alpha}$ regular. This can be thought of as a parabo...

We continue the study of Modica type gradient estimates for non-homogeneous parabolic equations initiated in \cite{BG}. First, we show that for the parabolic minimal surface equation with a semilinear force term if a certain gradient estimate is satisfied at $t=0$, then it holds for all later times $t>0$. We then establish analogous results for rea...

We continue the study of Modica type gradient estimates for inhomogeneous parabolic equations initiated in Banerjee and Garofalo (Nonlinear Anal. Theory Appl., to appear). First, we show that for the parabolic minimal surface equation with a semilinear force term if a certain gradient estimate is satisfied at t = 0, then it holds for all later time...

In this paper, we study an inhomogeneous variant of the normalized $p$-Laplacian evolution which has been recently treated in \cite{BG1}, \cite{Do}, \cite{MPR} and \cite{Ju}. We show that if the initial datum satisfies the pointwise gradient estimate \eqref{e:main1} a.e., then the unique solution to the Cauchy problem \eqref{main5} satisfies the sa...

In this paper, we study the potential theoretic aspects of the normalized p-Laplacian evolution, see (1.1) below. A systematic study of such equation was recently started in [1], [4] and [25]. Via the classical Perron approach, we address the question of solvability of the Cauchy-Dirichlet problem with "very weak" assumptions on the boundary of the...

We establish the strong unique continuation property for solutions to where F satisfies the structural assumptions A)-D). This extends a recent result of Armstrong and Silvestre (see) where F was assumed to be independent of x. We also establish an analogous unique continuation result at the boundary along the lines of when the domain is C3,α.

Let $ \ti \Om $ be a bounded convex domain in Euclidean $ n $ space, $ \hat x \in \ar \ti \Om, $ and $ r > 0. $ Let $ \ti u = (\ti u^1, \ti u^2, \dots, \ti u^N) $ be a weak solution to \[\nabla \cdot \left (|\nabla \ti u |^{p-2} \nabla \ti u \right) = 0 \mbox{in} \ti \Om \cap B (\hat x, 4 r) \mbox{with} |\nabla \ti u|^{p-2} \, \ti u_\nu = 0 \mbox{o...

We make the observation that, under some natural conditions on FF (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)(x0,t0), then there is a small spatial neighborhood Br0(x0)×{t0}Br0(x0)×{t0} of (x0,t0)(x0,t0) in which uu vanishes identicall...

In this paper we consider fully nonlinear parabolic equations in (Formula presented.) of the type(Formula presented.)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 to the differential inequalities in (2.5) below. Such inequalities inv...

We construct viscosity solutions to the nonlinear evolution equation \eqref{p} below which generalizes the motion of level sets by mean curvature (the latter corresponds to the case $p = 1$) using the regularization scheme as in \cite{ES1} and \cite{SZ}. The pointwise properties of such solutions, namely the comparison principles, convergence of so...