1,048 research outputs found
Modica Type Gradient Estimates for Reaction-Diffusion Equations
No full textWe 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 times t > 0. We then establish analogous results for reaction-diffusion equations such as (5) below in Ω × [0, T], where Ω is an epigraph such that the mean curvature of ∂ Ω is nonnegative. We then turn our attention to settings where such gradient estimates are valid without any a priori information on whether the estimate holds at some earlier time. Quite remarkably (see Theorems 4.1, 4.2 and 5.1), this is true for Rn×(−∞,0] and Ω×(−∞,0], where Ω is an epigraph satisfying the geometric assumption mentioned above, and for M×(−∞,0], where M is a connected, compact Riemannian manifold with nonnegative Ricci tensor. As a consequence of the gradient estimate (7), we establish a rigidity result (see Theorem 6.1 below) for solutions to (5) which is the analogue of Theorem 5.1 in Caffarelli et al. (Commun. Pure Appl. Math. 47, 1457–1473, 1994). Finally, motivated by Theorem 6.1, we close the paper by proposing a parabolic version of the famous conjecture of De Giorgi also known as the ε-version of the Bernstein theorem
Gradient bounds and monotonicity of the energy for some nonlinear singular diffusion equations
No full textQuantitative uniqueness for zero-order perturbations of generalized Baouendi-Grushin operators
No full textBased 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 unique continuation that can be thought of as an analogue
of the recent results of Bakri and Zhu for the standard Laplacian
Quantitative uniqueness for elliptic equations at the boundary of C1,Dini domains
No full textBased 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 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 by Bakri and Zhu for the standard Laplacian
Monotonicity of generalized frequencies and the strong unique continuation property for fractional parabolic equations
No full textA parabolic analogue of the higher-order comparison theorem of De Silva and Savin
No full textWe show that the quotient of two caloric functions which vanish on a portion of the lateral boundary of a Hk+αHk+α domain is Hk+αHk+α up to the boundary for k≥2k≥2. In the case k=1k=1, we show that the quotient is in H1+αH1+α if the domain is assumed to be space-time C1,αC1,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 give counterexamples to the fact that analogous results are not true at points on the parabolic boundary which are not on the lateral boundary, i.e., points which are at the corner and base of the parabolic boundary
Modica type gradient estimates for an inhomogeneous variant of the normalized p-Laplacian evolution
No full texta
A Stacked Segmented Adaptive Power Amplifier in 22nm FD-SOI
No full textThis work was supported by Soitec. (Corresponding author: Aritra Banerjee.
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