7 research outputs found
Hölder continuity and Harnack estimate for non-homogeneous parabolic equations
No full textIn this paper we continue the study on intrinsic Harnack inequality for non-homogeneous parabolic equations in non-divergence form initiated by the first author in Arya (Calc Var Partial Differ Equ 61:30–31, 2022). We establish a forward-in-time intrinsic Harnack inequality, which in particular implies the Hölder continuity of the solutions. We also provide a Harnack type estimate on global scale which quantifies the strong minimum principle. In the time-independent setting, this together with Arya (2022) provides an alternative proof of the generalized Harnack inequality proven by the second author in Julin (Arch Ration Mech Anal 216:673–702, 2015).peerReviewe
H\"older Continuity and Harnack estimate for non-homogeneous parabolic equations
Get PDFIn this paper we continue the study on intrinsic Harnack inequality for non-
homogeneous parabolic equations in non-divergence form initiated by the first
author in [1]. We establish a forward-in-time intrinsic Harnack inequality,
which in particular implies the H\"older continuity of the solutions. We also
provide a Harnack type estimate on global scale which quantifies the strong
minimum principle. In the time-independent setting, this together with [1]
provides an alternative proof of the generalized Harnack inequality proven by
the second author in [9]
Carleman estimates for sub-Laplacians on Carnot groups
No full textIn this note, we establish a new Carleman estimate with singular weights for the sub-Laplacian on a Carnot group G
for functions satisfying the discrepancy assumption in (2.16) below. We use such an estimate to derive a sharp vanishing order estimate for solutions to stationary Schrödinger equations.peerReviewe
Space-like quantitative uniqueness for parabolic operators
No full textWe obtain sharp maximal vanishing order at a given time level for solutions
to parabolic equations with a potential . 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 instead averaged over time. The principal
tool in our analysis is a new quantitative version of the well-known
Escauriaza-Fernandez-Vessella type Carleman estimate that we establish in our
setting.Comment: a revised versio
Optimal regularity for the variable coefficients parabolic Signorini problem
Get PDFIn this paper we discuss the optimal regularity of the variable coefficient parabolic Signorini problem with coefficients and inhomogeneity, where with being the space dimension. Relying on an parabolic Carleman estimate and an epiperimetric inequality, we show the optimal regularity of the solutions as well as the regularity of the regular free boundary
Sharp order of vanishing for parabolic equations, nodal set estimates and Landis type results
Get PDFWe 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 solutions
The asymptotic of the Mullins-Sekerka and the area-preserving curvature flow in the planar flat torus
No full textWe study the asymptotic behavior of flat flow solutions to the periodic and planar two-phase Mullins-Sekerka flow and area-preserving curvature flow. We show that flat flows converge to either a finite union of equally sized disjoint disks or to a finite union of disjoint strips or to the complement of these configurations exponentially fast. A key ingredient in our approach is the derivation of a sharp quantitative Alexandrov inequality for periodic smooth sets
