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Naian Liao

Publications and source records attributed to Naian Liao.

30 records · Page 2Linked to original sources
arXiv2021

On the Hölder regularity of signed solutions to a doubly nonlinear equation. Part II

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 \[ \partial_t\big(|u|^{q-1}u\big)-Δ_p u=0,\quad p>2,\quad 0<q<p-1. \] The first proof takes advantage of the expansion of positivity for the degenerate, parabolic $p$-Laplacian, thus simplifying the argument; whereas the other proof relies solely on the energy estimates for the doubly nonlinear parabolic equations. After proper adaptions of the interior arguments, we also obtain the boundary regularity for initial-boundary value problems of Dirichlet type and Neumann type.

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arXiv2021

Local continuity of weak solutions to the Stefan problem involving the singular $p$-Laplacian

We establish the local continuity of locally bounded weak solutions (temperatures) to the doubly singular parabolic equation modeling the phase transition of a material: \[ \partial_t β(u)-Δ_p u\ni 0\quad\text{ for }\tfrac{2N}{N+1}<p<2, \] where $β$ is a maximal monotone graph with a jump at zero and $Δ_p$ is the $p$-Laplacian. Moreover, a logarithmic type modulus of continuity is quantified, which has been conjectured to be optimal.

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arXiv2021

On the logarithmic type boundary modulus of continuity for the Stefan problem

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 Dirichlet problem, we merely assume that the spatial domain satisfies a measure density property, and the boundary datum has a logarithmic type modulus of continuity. For the Neumann problem, we assume that the lateral boundary is smooth, and the boundary datum is bounded. The proofs are measure theoretical in nature, exploiting De Giorgi's iteration and refining DiBenedetto's approach. Based on the sharp quantitative estimates, construction of continuous weak (physical) solutions is also indicated. The logarithmic type modulus of continuity has been conjectured to be optimal as a structural property for weak solutions to such partial differential equations.

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arXiv2021

Regularity of weak supersolutions to elliptic and parabolic equations: lower semicontinuity and pointwise behavior

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 weak supersolutions become lower semicontinuous after redefinition on a set of measure zero. The proof relies on a general principle, i.e. the De Giorgi type lemma, which offers a unified approach for a wide class of elliptic and parabolic equations, including an anisotropic elliptic equation, the parabolic $p$-Laplace equation, and the porous medium equation. In the second part, we shall show that for parabolic problems the lower semicontinuous representative at an instant can be recovered pointwise from the "essliminf" of past times. We also show that it can be recovered by the limit of certain integral average of past times. The proof hinges on the expansion of positivity for weak supersolutions. Our results are structural properties of partial differential equations, independent of any kind of comparison principle.

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arXiv2020

A Boundary Estimate for Singular Sub-Critical Parabolic Equations

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 $p$-laplacian type, with $p$ in the sub-critical range $(1,\frac{2N}{N+1}]$. The estimate is given in terms of a Wiener-type integral, defined by a proper elliptic $p$-capacity.

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arXiv2020

Remarks on parabolic De Giorgi classes

We make several remarks concerning properties of functions in parabolic De Giorgi classes of order $p$. There are new perspectives including a novel mechanism of propagating positivity in measure, the reservation of membership under convex composition, and a logarithmic type estimate. Based on them, we are able to give new proofs of known properties. In particular, we prove local boundedness and local Hölder continuity of these functions via Moser's ideas, thus avoiding De Giorgi's heavy machinery. We also seize this opportunity to give a transparent proof of a weak Harnack inequality for non-negative members of some super-class of De Giorgi, without any covering argument.

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arXiv2020

On the Hölder regularity of signed solutions to a doubly nonlinear equation

We establish the interior and boundary Hölder continuity of possibly sign-changing solutions to a class of doubly nonlinear parabolic equations whose prototype is \[ \partial_t\big(|u|^{p-2}u\big)-Δ_p u=0,\quad p>1. \] The proof relies on the property of expansion of positivity and the method of intrinsic scaling, all of which are realized by De Giorgi's iteration. Our approach, while emphasizing the distinct roles of sub(super)-solutions, is flexible enough to obtain the Hölder regularity of solutions to initial-boundary value problems of Dirichlet type or Neumann type in a cylindrical domain, up to the parabolic boundary. In addition, based on the expansion of positivity, we are able to give an alternative proof of Harnack's inequality for non-negative solutions. Moreover, as a consequence of the interior estimates, we also obtain a Liouville-type result.

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arXiv2018

A Boundary Estimate for Degenerate Parabolic Diffusion Equations

We prove an estimate on the modulus of continuity at a boundary point of a cylindrical domain for local weak solutions to degenerate parabolic equations of $p$-laplacian type. The estimate is given in terms of a Wiener-type integral, defined by a proper elliptic $p$-capacity.

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arXiv2017

A Boundary Estimate for Singular Parabolic Diffusion Equations

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 p-laplacian type. The estimate is given in terms of a Wiener-type integral, defined by a proper elliptic p-capacity.

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arXiv2013

Two Remarks on the Local Behavior of Solutions to Logarithmically Singular Diffusion Equations and its Porous-Medium Type Approximations

For the logarithmically singular parabolic equation \[ u_t-Δ\ln u=0\qquad\text{weakly in}\ \ E\times(0,T], \] we establish a Harnack type estimate in the $L^1_{loc}$ topology, and we show that the solutions are locally analytic in the space variables and differentiable in time. The main assumption is that $\ln u$ possesses a sufficiently high degree of integrability, namely \begin{equation*} \ln u\in L^\infty_{loc}\big(0,T;L^p_{loc}(E)\big) \quad\text{for some} p\ge1. \end{equation*} These two properties are known for solutions of singular porous medium type equations ($0<m<1$), which formally approximate the logarithmically singular equation. However, the corresponding estimates deteriorate as $m\to0$. It is shown that these estimates become stable and carry to the limit as $m\to0$, provided the indicated sufficiently high order of integrability is in force. The latter then appears as the discriminating assumption between solutions of parabolic equations with power-like singularities and logarithmic singularities to insure such solutions to be regular.

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