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15

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309

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Citations since 2016

## Publications

Publications (15)

In this paper, we consider the initial and boundary value problem of Ericksen-Leslie system modeling nematic liquid crystal flows in dimension three. Two examples of singularity at finite time are constructed. The first example is constructed in a special axisymmetric class with suitable axisymmetric initial and boundary data, while the second exam...

A singularly perturbed free boundary problem arising from a real problem associated with a Radiographic Integrated Test Stand concerns a solution of the equation $\Delta u = f(u)$ in a domain $\Omega$ subject to constant boundary data, where the function $f$ in general is not monotone. When the domain $\Omega$ is a perfect ring, we incorporate a ne...

A bifurcation about the uniqueness of a solution of a singularly perturbed free boundary problem of phase transition associated with the p-Laplacian, subject to given boundary condition is proved in this paper. We show this phenomenon by proving the existence of a third solution through the Mountain Pass Lemma when the boundary data decreases below...

In this paper, we study the limit as p goes to infinity of a minimizer of a variational problem that is a two-phase free boundary problem of phase transition for the p-Laplacian. Under a geometric compatibility condition, we prove that this limit is a solution of a free boundary problem for the infinity-Laplacian. When the compatibility condition d...

In this paper, we prove that a bifurcation phenomenon exists in a one-phase singularly perturbed free boundary problem of phase transition. Namely, the uniqueness of a solution of the one-phase problem breaks down as the boundary data decreases through a threshold value. The minimizer of the functional in consideration separates from the trivial ha...

Liouville-type theorems are powerful tools in partial differential equations. Boundedness assumptions of solutions are often imposed in deriving such Liouville-type theorems. In this paper, we establish some Liouville-type theorems without the boundedness assumption of nonnegative solutions to certain classes of elliptic equations and systems. Usin...

In this paper, we explore a general method to derive Hp → Lp boundedness from Hp → Hp boundedness of linear operators, an idea originated in the work of Han and Lu in dealing with the multiparameter flag singular integrals ([19]). These linear operators include many singular integral operators in one parameter and multiparameter settings. In this p...

We analyze the set of continuous viscosity solutions of the infinity Laplace equation $-Delta^N_{infty}w(x) = f(x)$, with generally sign-changing right-hand side in a bounded domain. The existence of a least and a greatest continuous viscosity solutions, up to the boundary, is proved through a Perron's construction by means of a strict comparison p...

In this paper, we study the uniqueness problem of a two-phase elliptic free boundary problem arising from the phase transition problem subject to given boundary data. We show that in general the comparison principle between the sub- and super-solutions does not hold, and there is no uniqueness of either a viscosity solution or a minimizer of this f...

The inhomogeneous normalized infinity Laplace equation was derived from the tug-of-war game in [PSSW] with the positive right-hand-side as a running pay-off. The existence, uniqueness and comparison with polar quadratic functions were proved in [PSSW] by the game theory. In this paper, the normalized infinity Lapla-cian, formally written as N ∞ u =...

We present the theory of the viscosity solutions of the inhomogeneous infinity Laplace equation in domains in Rn. We show existence and uniqueness of a viscosity solution of the Dirichlet problem under the intrinsic condition f does not change its sign. We also discover a characteristic property, which we call the comparison with standard functions...

We consider the uniqueness of a viscosity solution of the degenerate elliptic equation F(x,u(x),∇u(x),D 2 u(x))=0 or its normalized counterpart F(x,u(x),∇u ^(x),D 2 u(x))=0 subject to continuous boundary data. A useful strict comparison principle is obtained with very weak requirements, precisely that F is proper, degenerate elliptic, and locally u...

Comparison results are obtained between infinity subharmonic and infinity superharmonic func-tions defined on unbounded domains. The primary new tool employed is an approximation of infinity subhar-monic functions that allows one to assume that gradients are bounded away from zero. This approximation also demystifies the proof in the case of a boun...

Sharp Poincaré inequalities on balls or chain type bounded domains have been extensively studied both in classical Euclidean space and Carnot–Carathéodory spaces associated with sub-elliptic vector fields (e.g., vector fields satisfying Hörmander's condition). In this paper, we investigate the validity of sharp global Poincaré inequalities of both...