arXiv, 2007 The Hopf Lemma for second order elliptic operators is proved to hold in domains with $C^{1,\alpha}$, and even less regular, boundaries. It need not hold for $C^1$ boundaries. Corresponding results are proved for second order parabolic operators.
arxiv
Singular limit of a two-phase flow problem in porous medium as the air viscosity tends to zero [PDF]
arXiv, 2009 In this paper we consider a two-phase flow problem in porous media and study its singular limit as the viscosity of the air tends to zero; more precisely, we prove the convergence of subsequences to solutions of a generalized Richards model.
arxiv
One-dimensional symmetry for solutions of Allen Cahn fully nonlinear equations [PDF]
arXiv, 2010 This article presents some qualitative results for entire solutions of the fully nonlinear elliptic equations of Allen Cahn type . Precisely under some additional assumptions on the forcing term, if the solution is bounded and converges uniformly at infinity in a fixed direction to its extrema, then the solution depends only on one variable.
arxiv
The Harnack inequality and related properties for solutions to elliptic and parabolic equations with divergence-free lower-order coefficients [PDF]
arXiv, 2010 We discuss the problem how "bad" may be lower-order coefficients in elliptic and parabolic second order equations to ensure some qualitative properties of solution such as strong maximum principle, Harnack's inequality, Liouville's theorem. The answers are given in terms of the Lebesgue spaces and the Morrey spaces.
arxiv
A centennial of the Zaremba--Hopf--Oleinik Lemma [PDF]
arXiv, 2010 We discuss the problem how "bad" may be lower-order coefficients in elliptic and parabolic second order equations to ensure the Hopf--Oleinik Lemma for solutions to hold true. We also touch the gradient estimates for solutions at the boundary.
arxiv
Maximum principle for semi-elliptic trace operators and geometric applications [PDF]
arXiv, 2012 Based on ideas of L. Al\'ias, D. Impera and M. Rigoli developed in "Hypersurfaces of constant higher order mean curvature in warped products", we develope a fairly general weak/Omori-Yau maximum principle for trace operators. We apply this version of maximum principle to generalize several higher order mean curvature estimates and to give an extension ...
arxiv
On the maximum principle for parabolic equations with unbounded coefficients [PDF]
arXiv, 2015 Translation of the paper "Interpolation of linear spaces and maximum estimates for solutions to parabolic equations" published in Russian in the collected volume "Partial differential equations", Akad. Nauk SSSR, Sibirsk. Otdel., Inst. Mat., Novosibirsk, 1987, 50--72. Original proofs are essentially simplified. Some gaps are fixed and some comments are
arxiv
Maximum principles for Laplacian and fractional Laplacian with critical integrability
, 2019 In this paper, we study maximum principles for Laplacian and fractional Laplacian with critical integrability. We first consider $-\Delta u(x)+c(x)u(x)\geq 0$ in $B_1$ where $c(x)\in L^{p}(B_1)$, $B_1\subset \mathbf{R}^n$. As is known that $p=\frac{n}{2}$
Lü, Yingshu
core
Rate of convergence for one-dimensional quasilinear parabolic problem and its applications [PDF]
arXiv, 2016 Based on a comparison principle, we derive an exponential rate of convergence for solutions to the initial-boundary value problem for a class of quasilinear parabolic equations in one space dimension. We then apply the result to some models in population dynamics and image processing.
arxiv
Symmetry and monotonicity properties of singular solutions to some cooperative semilinear elliptic systems involving critical nonlinearities [PDF]
arXiv, 2019 We investigate qualitative properties of positive singular solutions of some elliptic systems in bounded and unbounded domains. We deduce symmetry and monotonicity properties via the moving plane procedure. Moreover, in the unbounded case, we study some cooperative elliptic systems involving critical nonlinearities in $\mathbb{R}^n$.
arxiv