DISCONTINUOUS SOLUTIONS IN L^∞ FOR HAMILTON-JACOBI EQUATIONS

An approach is introduced to construct global discontinuous solutions in L∞ for Hamilton-Jacobi equations. This approach allows the initial data only in L∞ and applies to the equations with nonconvex Hamiltonians. The profit functions are introduced to formulate the notion of discontinuous solutions in L∞. The existence of global discontinuous solutions in L∞ is established. These solutions in L∞ coincide with the viscosity solutions and the minimax solutions, provided that the initial data are continuous. A prototypical equation is analyzed toexamine the L∞ stability of our L∞ solutions. The analysis also shows that global discontinuous solutions are determined by the topology in which the initial data are approximated.     

Asymptotics for Solutions of Elliptic Equations in Double Divergence Form

We consider weak solutions of an elliptic equation of the form ? _i ? _i (a _ij u) = 0 and their asymptotic properties at an interior point. We assume that the coefficients are bounded, measurable, complex-valued functions that stabilize as x → 0 in that the norm of the matrix (a _ij (x) ? δ _ij ) on the annulus B _2r \ B _r is bounded by a function Ω(r), where Ω²(r) satisfies the Dini condition at r = 0, as well as some technical monotonicity conditions; under these assumptions, solutions need not be continuous. Our main result is an explicit formula for the leading asymptotic term for solutions with at most a mild singularity at x = 0. As a consequence, we obtain upper and lower estimates for the L ^p -norm of solutions, as well as necessary and sufficient conditions for solutions to be bounded or tend to zero in L ^p -mean as r → 0.     

Qualitative Properties of Saddle-Shaped Solutions to Bistable Diffusion Equations

We consider the elliptic equation ? Δu = f(u) in the whole ?^2m , where f is of bistable type. It is known that there exists a saddle-shaped solution in ?^2m . This is a solution which changes sign in ?^2m and vanishes only on the Simons cone 𝒞 = {(x ¹, x ²) ∈ ? ^m × ? ^m : |x ¹| = |x ²|}. It is also known that these solutions are unstable in dimensions 2 and 4.

In this article we establish that when 2m = 6 every saddle-shaped solution is unstable outside of every compact set and, as a consequence has infinite Morse index. For this we establish the asymptotic behavior of saddle-shaped solutions at infinity. Moreover we prove the existence of a minimal and a maximal saddle-shaped solutions and derive monotonicity properties for the maximal solution.

These results are relevant in connection with a conjecture of De Giorgi on 1D symmetry of certain solutions. Saddle-shaped solutions are the simplest candidates, besides 1D solutions, to be global minimizers in high dimensions, a property not yet established.     

Lq‐estimates for the stationary Oseen equations on the exterior of a rotating obstacle

We study the Dirichlet problem for the stationary Oseen equations around a rotating body in an exterior domain. Our main results are the existence and uniqueness of weak and very weak solutions satisfying appropriate L^q‐estimates. The uniqueness of very weak solutions is shown by the method of cut‐off functions with an anisotropic decay. Then our existence result for very weak solutions is deduced by a duality argument from the existence and estimates of strong solutions. From this and interior regularity of very weak solutions, we finally establish the complete D^1,r‐result for weak solutions of the Oseen equations around a rotating body in an exterior domain, where 4/3<r <4. Here, D^1,r is the homogeneous Sobolev space.     

S-polar sets of super-brownian motions and solutions of nonlinear differential equations

This paper gives probabilistic expressions of the minimal and maximal positive solutions of the partial differential equation -1/2△v(x) γ(x)v(x)α = 0 in D, where D is a regular domain in Rd(d ≥ 3) such that its complement Dc is compact, γ(x) is a positive bounded integrable function in D, and 1 < α≤ 2. As an application, some necessary and sufficient conditions for a compact set to be S-polar are presented.     

Qualitative theory for strongly damped wave equations

We investigate the asymptotic periodicity, L^p‐boundedness, classical (resp., strong) solutions, and the topological structure of solutions set of strongly damped semilinear wave equations. The theoretical results are well complemented with a set of very illustrating applications.     

Non-Diffusive Large Time Behavior for a Degenerate Viscous Hamilton–Jacobi Equation

The convergence to non-diffusive self-similar solutions is investigated for non-negative solutions to the Cauchy problem ? _t u = Δ _p u + |? u| ^q when the initial data converge to zero at infinity. Sufficient conditions on the exponents p > 2 and q > 1 are given that guarantee that the diffusion becomes negligible for large times and the L ^∞-norm of u(t) converges to a positive value as t → ∞.     

Small perturbation solutions for nonlocal elliptic equations

We present a small perturbation result for nonlocal elliptic equations, which says that for a class of nonlocal operators, the solutions are in C^σ+α for any α∈(0,1) as long as the solutions are small. This is a nonlocal generalization of a celebrated result of Savin in the case of second order equations.     

Singly Periodic Solutions of the Allen-Cahn Equation and the Toda Lattice

The Allen-Cahn equation ? Δu = u ? u ³ in ?² has family of trivial singly periodic solutions that come from the one dimensional periodic solutions of the problem ?u″ =u ? u ³. In this paper we construct a non-trivial family of singly periodic solutions to the Allen-Cahn equation. Our construction relies on the connection between this equation and the infinite Toda lattice. We show that for each one-soliton solution to the infinite Toda lattice we can find a singly periodic solution to the Allen-Cahn equation, such that its level set is close to the scaled one-soliton. The solutions we construct are analogues of the family of Riemann minimal surfaces in ?³.     

Regularity for Fully Nonlinear Elliptic Equations with Neumann Boundary Data

We obtain local C ^α, C ^1,α, and C ^2,α regularity results up to the boundary for viscosity solutions of fully nonlinear uniformly elliptic second order equations with Neumann boundary conditions.     

Global periodicity and openness of the set of solutions for discrete dynamical systems

In this paper, we find additional conditions to be satisfied by a globally periodic discrete dynamical system, so that its good set (the set of initial conditions providing well-defined solutions) is an open set of ? ^k or ? ^k . We will pay especial attention to the rational case and several examples will be given.     

Boundary blow-up solutions for a class of quasilinear elliptic equations

For a given bounded domain Ω in R ^N with smooth boundary ?Ω, we give sufficient conditions on f so that the m-Laplacian equation △ _m u = f(x, u, ?u) admits a boundary blow-up solution u ∈ W ^1,p (Ω). Our main results are new and extend the results in J.V. Concalves and Angelo Roncalli [Boundary blow-up solutions for a class of elliptic equations on a bounded domain, Appl. Math. Comput. 182 (2006), pp. 13–23]. Our approach employs the method of lower–upper solution theorem, fixed point theory and weak comparison principle.     

Families of Periodic Solutions of Resonant PDEs

Summary. We construct some families of small amplitude periodic solutions close to a completely resonant equilibrium point of a semilinear reversible partial differential equation. To this end, we construct, using averaging methods, a suitable map from the configuration space to itself. We prove that to each nondegenerate zero of such a map there corresponds a family of small amplitude periodic solutions of the system. The proof is based on Lyapunov-Schmidt decomposition. This establishes a relation between Lyapunov-Schmidt decomposition and averaging theory that could be interesting in itself. As an application, we construct countable many families of periodic solutions of the nonlinear string equation u _tt -u _xx ± u ³ =0 (and of its perturbations) with Dirichlet boundary conditions. We also prove that the fundamental periods of solutions belonging to the n ^th family converge to 2π/n when the amplitude tends to zero. Received August 8, 2000; accepted November 21, 2000 Online publication February 26, 2001     

环域上一类拟线性椭圆型方程的正径向解的存在性

The existence of positive radial solutions of the equation -din( |Du|p-2Du)=f(u) is studied in annular domains in Rn,n≥2. It is proved that if f(0)≥0, f is somewherenegative in (0,∞), limu→0^ f‘ (u)=0 and limu→∞ (f(u)/u^p-1)=∞, then there is alarge positive radial solution on all annuli. If f(0)≤0 and satisfies certain conditions, then the equation has no radial solution if the annuli are too wide.     

Bifurcation for Quasilinear Elliptic Singular BVP

For a continuous function g ≥ 0 on (0, + ∞) (which may be singular at zero), we confront a quasilinear elliptic differential operator with natural growth in ?u, ? Δu + g(u)|?u|², with a power type nonlinearity, λu ^p  + f ₀(x). The range of values of the parameter λ for which the associated homogeneous Dirichlet boundary value problem admits positive solutions depends on the behavior of g and on the exponent p. Using bifurcations techniques we deduce sufficient conditions for the boundedness or unboundedness of the cited range.     

Uniqueness of Solutions on the Whole Time Axis to the Navier-Stokes Equations in Unbounded Domains

We consider the uniqueness of bounded continuous L^{3, ∞}-solutions on the whole time axis to the Navier-Stokes equations in 3-dimensional unbounded domains. Here, L^{p, q} denotes the scale of Lorentz spaces. Thus far, uniqueness of such solutions to the Navier-Stokes equations in unbounded domain, roughly speaking, is known only for a small solution in BC(?; L^{3, ∞}) within the class of solutions which have sufficiently small L^∞(L^{3, ∞})-norm. In this paper, we discuss another type of uniqueness theorem for solutions in BC(?; L^{3, ∞}) using a smallness condition for one solution and a precompact range condition for the other one. The proof is based on the method of dual equations.     

Asymptotic behaviour of global classical solutions of diagonalizable quasilinear hyperbolic systems

This paper is concerned with the asymptotic behaviour of global classical solutions of diagonalizable quasilinear hyperbolic systems with linearly degenerate characteristic fields. Based on the existence results of global classical solutions, we prove that when t tends to infinity, the solution approaches a combination of C¹ travelling wave solutions, provided that L¹ ∩ L^∞ norm of the initial data as well as its derivative are bounded. Application is given for the time‐like extremal surface in Minkowski space. Copyright © 2006 John Wiley & Sons, Ltd.     

Symmetry Properties of Positive Solutions of Parabolic Equations on ℝ N : I. Asymptotic Symmetry for the Cauchy Problem

We consider quasilinear parabolic equations on ? ^N satisfying certain symmetry conditions. We prove that bounded positive solutions decaying to zero at spatial infinity are asymptotically radially symmetric about a center. The asymptotic center of symmetry is not fixed a priori (and depends on the solution) but it is independent of time. We also prove a similar theorem on reflectional symmetry.     

Global strong solutions and time decay of 2D tropical climate model with zero thermal diffusion

The global small solutions of the tropical climate model are obtained with the fractional dissipative terms Λ^αu in the equation of the barotropic mode u and Λ^αv in the equation of the first baroclinic mode v. More precisely, we prove for 1<α ≤ 2 that the couple system has global unique strong solutions for small initial data with critical regularities. Moreover, the smallness assumption imposed on the initial barotropic mode of the velocity can be removed if α=2. We also study the large time behavior of the constructed solutions and obtain optimal time decay rates by a pure energy argument.     

On the uniqueness of time-periodic solutions to the Navier–Stokes equations in unbounded domains

We present a uniqueness theorem for time-periodic solutions to the Navier–Stokes equations in unbounded domains. Thus far, results on the uniqueness of time-periodic solutions to the Navier–Stokes equations in unbounded domain, roughly speaking, have only found that a small time-periodic L ⁿ -solution is unique within the class of solutions which have sufficiently small L ^∞(L ⁿ )-norm. In this paper, we show that a small time-periodic L ⁿ -solution is unique within the class of all time-periodic L ⁿ -solutions, which contains large solutions. We also consider the uniqueness of solutions in weak-L ⁿ space. The proof of the present uniqueness theorem is based on the method of dual equations.