Existence and estimates of solutions to a singular Dirichlet problem for the Monge–Ampère equation

Given a strictly convex, smooth, and bounded domain Ω in we establish the existence of a negative convex solution in with zero boundary value to the singular Monge–Ampère equation det(D²u)=p(x)g(−u). An associated Dirichlet problem will be employed to provide a necessary and sufficient condition for the solvability of the singular boundary value problem. Estimates of solutions will also be given and regularity of solutions will be deduced from the estimates.     

New isoperimetric estimates for solutions to Monge–Ampère equations

We prove some sharp estimates for solutions to Dirichlet problems relative to Monge–Ampère equations. Among them we show that the eigenvalue of the Dirichlet problem, when computed on convex domains with fixed measure, is maximal on ellipsoids. This result falls in the class of affine isoperimetric inequalities and shows that the eigenvalue of the Monge–Ampère operator behaves just the contrary of the first eigenvalue of the Laplace operator.     

A logarithmic Gauss curvature flow and the Minkowski problem

Let X₀ be a smooth uniformly convex hypersurface and f a postive smooth function in Sⁿ. We study the motion of convex hypersurfaces X(·,t) with initial X(·,0)=θX₀ along its inner normal at a rate equal to log(K/f) where K is the Gauss curvature of X(·,t). We show that the hypersurfaces remain smooth and uniformly convex, and there exists θ^*>0 such that if θ<θ^*, they shrink to a point in finite time and, if θ>θ^*, they expand to an asymptotic sphere. Finally, when θ=θ^*, they converge to a convex hypersurface of which Gauss curvature is given explicitly by a function depending on f(x).     

Finitely smooth solutions of nonlinear singular partial differential equations

We solve a Fuchsian system of singular nonlinear partial differential equations with resonances. These equations have no smooth solutions in general. We show the solvability in a class of finitely smooth functions. Typical examples are a homology equation for a vector field and a degenerate Monge–Ampère equation. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Interior gradient estimates for solutions to the linearized Monge–Ampère equation

Let ? be a convex function on a convex domain Ω⊂Rn, n?1. The corresponding linearized Monge–Ampère equation istrace(ΦD²u)=f, where is the matrix of cofactors of D²?. We establish interior Hölder estimates for derivatives of solutions to such equation when the function f on the right-hand side belongs to Lp(Ω) for some p>n. The function ? is assumed to be such that with ?=0 on ∂Ω and the Monge–Ampère measure is given by a density g∈C(Ω) which is bounded away from zero and infinity.     

The domain of definition of the quaternionic Monge–Ampère operator

In this paper, we give an equivalent characterization of the domain of definition for the quaternionic Monge–Ampère operator, by using the theory of quaternionic closed positive current we established in [17–19]. This domain of definition for the complex Monge–Ampère operator was introduced by Blocki [7,8].     

Radially symmetric convex solutions for Dirichlet problems of Monge‐Ampère equations

In this paper, we study the existence of radially symmetric convex solutions for Dirichlet problems of Monge‐Ampère equations. By applying a well‐known fixed point theorem in cones, we shall establish several new criteria for the existence of nontrivial radially symmetric convex solutions for the systems of Monge‐Ampère equations with or without an eigenvalue parameter. Copyright © 2015 John Wiley & Sons, Ltd.     

Partial pluricomplex energy and integrability exponents of plurisubharmonic functions

We first prove a quantitative estimate of the volume of the sublevel sets of a plurisubharmonic function in a hyperconvex domain with boundary values 0 (in a quite general sense) in terms of its Monge–Ampère mass in the domain. Then we deduce a sharp sufficient condition on the Monge–Ampère mass of such a plurisubharmonic function φ for exp(−2φ) to be globally integrable as well as locally integrable.     

Periodic solutions to the Cahn–Hilliard equation with constraint

This paper is concerned with the multidimensional Cahn–Hilliard equation with a constraint. The existence of periodic solutions of the problem is mainly proved under consideration by the viscosity approach. More precisely, with the help of the subdifferential operator theory and Schauder fixed point theorem, the existence of solutions to the approximation of the original problem is shown, and then the solution is obtained by using a passage‐to‐limit procedure based on a prior estimate. Copyright © 2015 John Wiley & Sons, Ltd.     

Global well‐posedness of a damped Schrödinger equation in two space dimensions

We investigate the initial value problem for a semilinear damped Schrödinger equation with exponential growth nonlinearity in two space dimensions. We obtain global well‐posedness in the energy space. Copyright © 2013 John Wiley & Sons, Ltd.     

The local regularity for strong solutions of the Hessian quotient equation

By means of the Reilly formula and the Alexandrov maximum principle, we obtain the local C1,1 estimates of the W2,p strong solutions to the Hessian quotient equations for p sufficiently large, and then prove that these solutions are smooth. There are counterexamples to show that the integral exponent p is optimal in some cases. We modify partially the known result in the Hessian case, and extend the regularity result in the special Lagrangian case to the Hessian quotient case.     

On the fractional Newton and wave equation in one space dimension

We consider the new mathematical scenarios in the framework of the Fractional Calculus. In this context, we study the generalized fractional virial theorem as well as the fractional plane wave solutions and the fractional dispersion relations for the fractional wave equation.     

The Cauchy problem for the homogeneous Monge–Ampère equation, II. Legendre transform

We continue our study of the Cauchy problem for the homogeneous (real and complex) Monge–Ampère equation (HRMA/HCMA). In the prequel (Y.A. Rubinstein and S. Zelditch [27]) a quantum mechanical approach for solving the HCMA was developed, and was shown to coincide with the well-known Legendre transform approach in the case of the HRMA. In this article—that uses tools of convex analysis and can be read independently—we prove that the candidate solution produced by these methods ceases to solve the HRMA, even in a weak sense, as soon as it ceases to be differentiable. At the same time, we show that it does solve the equation on its dense regular locus, and we derive an explicit a priori upper bound on its Monge–Ampère mass. The technique involves studying regularity of Legendre transforms of families of non-convex functions.     

A study of the solutions of the combined sine–cosine-Gordon equation

We have studied the solutions of the combined sine–cosine-Gordon Equation found by Wazwaz [A.M. Wazwaz, Travelling wave solutions for combined and double combined sine-cosine-Gordon equations by the variable separated ODE method, Appl. Math. Comput. 177 (2006) 755] using the variable separated ODE method. These solutions can be transformed into a new form. We have derived the relation between the phase of the combined sine–cosine-Gordon equation and the parameter in these solutions. Its applications in physical systems are also discussed.