11 citations · 41 across the 27 of their papers we have counts for
29 papers
Nonlocal intrinsic Harnack estimates
Naian Liao
We extend DiBenedetto's intrinsic parabolic Harnack estimates to a class of integro-differential equations with measurable kernels modeled on the fractional -Laplacian. More rem…
Sharp gradient integrability for -Poisson type equations
Verena Bögelein, Frank Duzaar, Naian Liao +1
We prove local -regularity for weak solutions to fractional -Laplacian type equations with right-hand side . Assuming , , an…
Regularity theory for sub-critical -parabolic systems with measurable coefficients
Verena Bögelein, Frank Duzaar, Ugo Gianazza +1
A quantitative regularity theory is developed for weak solutions to the parabolic system $$ \partial_t u-\mathrm{div}\,{\boldsymbol{\mathsf A}}(x,t,Du)=0 \quad\text{in }E_T\subset…
Nonnegative solutions to nonlocal parabolic equations
Naian Liao, Marvin Weidner
We aim to study nonnegative, global solutions to a general class of nonlocal parabolic equations with bounded measurable coefficients. First, we prove a Widder-type theorem. Such a…
Gradient estimates for the fractional -Poisson equation
Verena Bögelein, Frank Duzaar, Naian Liao +1
We consider local weak solutions to the fractional -Poisson equation of order , i.e. . In the range and we…
Local boundedness and higher integrability for the sub-critical singular porous medium system
Verena Bögelein, Frank Duzaar, Ugo Gianazza +1
The gradient of weak solutions to porous medium-type equations or systems possesses a higher integrability than the one assumed in the pure notion of a solution. We settle the crit…
Gradient regularity for -harmonic functions
Verena Bögelein, Frank Duzaar, Naian Liao +2
We study the local regularity properties of -harmonic functions, i.e. local weak solutions to the fractional -Laplace equation of order in the case $p\in (1,…
Time-insensitive nonlocal parabolic Harnack estimates
Naian Liao, Marvin Weidner
We establish new Harnack estimates that defy the waiting-time phenomenon for global solutions to nonlocal parabolic equations. Our technique allows us to consider general nonlocal…
Improved moduli of continuity for degenerate phase transitions
Ugo Gianazza, Naian Liao, José Miguel Urbano
We substantially improve in two scenarios the current state-of-the-art modulus of continuity for weak solutions to the -dimensional, two-phase Stefan problem featuring a deg…
Regularity for the fractional -Laplace equation
Verena Bögelein, Frank Duzaar, Naian Liao +2
Higher Sobolev and Hölder regularity is studied for local weak solutions of the fractional -Laplace equation of order in the case . Depending on the regime considere…
Harnack estimates for nonlocal drift-diffusion equations
Naian Liao
A set of pointwise estimates are established for local solutions to nonlocal diffusion equations with a drift term. In particular, our Harnack estimates are the first ones for such…
On the modulus of continuity of solutions to nonlocal parabolic equations
Naian Liao
A general modulus of continuity is quantified for locally bounded, local, weak solutions to nonlocal parabolic equations, under a minimal tail condition. Hölder modulus of continui…
Hölder Continuity of the Gradient of Solutions to Doubly Non-Linear Parabolic Equations
Verena Bögelein, Frank Duzaar, Ugo Gianazza +2
This paper is devoted to studying the local behavior of non-negative weak solutions to the doubly non-linear parabolic equation \begin{equation*} \partial_t u^q - \text{div}\big(|D…
Hölder regularity for parabolic fractional -Laplacian
Naian Liao
Local Hölder regularity is established for certain weak solutions to a class of parabolic fractional -Laplace equations with merely measurable kernels. The proof uses DeGiorgi's…
Boundary regularity for parabolic systems in convex domains
Verena Bögelein, Frank Duzaar, Naian Liao +1
In a cylindrical space-time domain with a convex, spatial base, we establish a local Lipschitz estimate for weak solutions to parabolic systems with Uhlenbeck structure up to the l…
Continuity of the temperature in a multi-phase transition problem
Ugo Gianazza, Naian Liao
Locally bounded, local weak solutions to a doubly nonlinear parabolic equation, which models the multi-phase transition of a material, is shown to be locally continuous. Moreover,…
On the Hölder regularity of signed solutions to a doubly nonlinear equation. Part III
Naian Liao, Leah Schätzler
We establish the local Hölder continuity of possibly sign-changing solutions to a class of doubly nonlinear parabolic equations whose prototype is \[ \partial_t\big(|u|^{q-1}u\big)…
On the Hölder regularity of signed solutions to a doubly nonlinear equation. Part II
Verena Bögelein, Frank Duzaar, Naian Liao +1
We demonstrate two proofs for the local Hölder continuity of possibly sign-changing solutions to a class of doubly nonlinear parabolic equations whose prototype is \[ \partial_t\bi…
Local continuity of weak solutions to the Stefan problem involving the singular -Laplacian
Naian Liao
We establish the local continuity of locally bounded weak solutions (temperatures) to the doubly singular parabolic equation modeling the phase transition of a material: \[ \partia…
On the logarithmic type boundary modulus of continuity for the Stefan problem
Naian Liao
A logarithmic type modulus of continuity is established for weak solutions to a two-phase Stefan problem, up to the parabolic boundary of a cylindrical space-time domain. For the D…
Regularity of weak supersolutions to elliptic and parabolic equations: lower semicontinuity and pointwise behavior
Naian Liao
We demonstrate a measure theoretical approach to the local regularity of weak supersolutions to elliptic and parabolic equations in divergence form. In the first part, we show that…
Remarks on parabolic De Giorgi classes
Naian Liao
We make several remarks concerning properties of functions in parabolic De Giorgi classes of order . There are new perspectives including a novel mechanism of propagating positi…
On the Hölder regularity of signed solutions to a doubly nonlinear equation
Verena Bögelein, Frank Duzaar, Naian Liao
We establish the interior and boundary Hölder continuity of possibly sign-changing solutions to a class of doubly nonlinear parabolic equations whose prototype is \[ \partial_t\big…
A Boundary Estimate for Singular Sub-Critical Parabolic Equations
Ugo Gianazza, Naian Liao
We prove an estimate on the modulus of continuity at a boundary point of a cylindrical domain for local weak solutions to singular parabolic equations of -laplacian type, with $…