Multiple solutions of nonlinear elliptic systems

We proved a multiplicity result for a nonlinear elliptic system in R^N. The functional related to the system is strongly indefinite. We investigated the relation between the number of solutions and the topology of the set of the global maxima of the coefficients.     

Single-point condensation phenomena for a four-dimensional biharmonic Ren–Wei problem

We continue to study the asymptotic behavior of least energy solutions to the following fourth order elliptic problem (E _p ): as p gets large, where Ω is a smooth bounded domain in R ⁴ . In our earlier paper (Takahashi in Osaka J. Math., 2006), we have shown that the least energy solutions remain bounded uniformly in p and they have one or two “peaks” away form the boundary. In this note, following the arguments in Adimurthi and Grossi (Proc. AMS 132(4):1013–1019, 2003) and Lin and Wei (Comm. Pure Appl. Math. 56:784–809, 2003), we will obtain more sharper estimates of the upper bound of the least energy solutions and prove that the least energy solutions must develop single-point spiky pattern, under the assumption that the domain is convex.     

Existence, uniqueness and asymptotic stability of time periodic traveling waves for a periodic Lotka–Volterra competition system with diffusion

We study the existence, uniqueness, and asymptotic stability of time periodic traveling wave solutions to a periodic diffusive Lotka–Volterra competition system. Under certain conditions, we prove that there exists a maximal wave speed c^? such that for each wave speed c?c^?, there is a time periodic traveling wave connecting two semi-trivial periodic solutions of the corresponding kinetic system. It is shown that such a traveling wave is unique modulo translation and is monotone with respect to its co-moving frame coordinate. We also show that the traveling wave solutions with wave speed c<c^? are asymptotically stable in certain sense. In addition, we establish the nonexistence of time periodic traveling waves for nonzero speed c>c^?.     

Multiple Positive Solutions for Eigenvalue Problems of Hemivariational Inequalities

We study a nonlinear eigenvalue problem with a nonsmooth potential. The subgradients of the potential are only positive near the origin (from above) and near +∞. Also the subdifferential is not necessarily monotone (i.e. the potential is not convex). Using variational techniques and the method of upper and lower solutions, we establish the existence of at least two strictly positive smooth solutions for all the parameters in an interval. Our approach uses the nonsmooth critical point theory for locally Lipschitz functions. A byproduct of our analysis is a generalization of a result of Brezis-Nirenberg (CRAS, 317 (1993)) on H¹₀ versus C¹₀ minimizers of a C¹-functional.     

On the number of nodal solutions to a singularly perturbed Neumann problem

We show that for ε small, there are arbitrarily many nodal solutions for the following nonlinear elliptic Neumann problem where Ω is a bounded and smooth domain in ℝ² and f grows superlinearly. (A typical f(u) is f(u)= a₁ u₊^p – a₁ u_-^p, a₁, a₂ >0, p, q>1.) More precisely, for any positive integer K, there exists ε_K>0 such that for 0<ε<ε_K, the above problem has a nodal solution with K positive local maximum points and K negative local minimum points. This solution has at least K+1 nodal domains. The locations of the maximum and minimum points are related to the mean curvature on ∂Ω. The solutions are constructed as critical points of some finite dimensional reduced energy functional. No assumption on the symmetry, nor the geometry, nor the topology of the domain is needed.     

Necessity of a Wiener type condition for boundary regularity of quasiminimizers and nonlinear elliptic equations

We use variational methods to obtain a pointwise estimate near a boundary point for quasisubminimizers of the p-energy integral and other integral functionals in doubling metric measure spaces admitting a p-Poincaré inequality. It implies a Wiener type condition necessary for boundary regularity for p-harmonic functions on metric spaces, as well as for (quasi)minimizers of various integral functionals and solutions of nonlinear elliptic equations on R ⁿ .     

Optimal <Emphasis Type="Italic">W</Emphasis><Superscript>1,<Emphasis Type="Italic">p</Emphasis></Superscript> regularity theory for parabolic equations in divergence form

This work treats L^p regularity theory for weak solutions of parabolic equations in divergence form with discontinuous coefficients on nonsmooth domains. We essentially obtain an optimal condition on the coefficients under which the global W^1,p regularity theory holds. This work was supported by SNU foundation in 2005.     

Convergence of the Lax–Friedrichs scheme for isothermal gas dynamics with semiconductor devices

In this paper we study the existence of global solutions to the Euler equations of compressible isothermal gas dynamics with semiconductor devices. We construct the approximate solutions by Lax–Friedrichs scheme. The convergence and consistency are obtained by using the compensated compactness framework for γ = 1. The global entropy solutions in L^∞ are obtained. We deal with the initial data containing unbounded velocity which is different from the isentropic case. Received: November 18, 2003     

Convergence of an implicit, constraint preserving finite element discretization of p-harmonic heat flow into spheres

We propose an implicit discretization of the p-harmonic map heat flow into the sphere S ² that enjoys a discrete energy inequality and converges under only a mild mesh constraint to a weak solution. A fully practical iterative scheme that approximates the solution of the nonlinear system of equations in each time step is proposed and analyzed. Computational studies to motivate possible finite-time blow-up behavior of solutions for p ≠ 2 are included. Supported by Deutsche Forschungsgemeinschaft through the DFG Research Center Matheon “Mathematics for key technologies” in Berlin.     

Maximum principle and existence of <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-viscosity solutions for fully nonlinear uniformly elliptic equations with measurable and quadratic terms

We study L^p-viscosity solutions of fully nonlinear, second-order, uniformly elliptic partial differential equations (PDE) with measurable terms and quadratic nonlinearity. We present a sufficient condition under which the maximum principle holds for L^p-viscosity solution. We also prove stability and existence results for the equations under consideration.     

Error estimate for finite volume scheme

We study the convergence of a finite volume scheme for the linear advection equation with a Lipschitz divergence-free speed in R ^d . We prove a h ^1/2-error estimate in the L ^∞(0,t;L ¹)-norm for BV data. This result was expected from numerical experiments and is optimal.     

Kernel estimates for Schrödinger operators

We prove short time estimates for the heat kernel of Schr?dinger operators with unbounded potential in R^N.     

<Emphasis Type="Bold">C</Emphasis><Superscript><Emphasis Type="Bold">1,</Emphasis><Emphasis Type="Italic">α</Emphasis></Superscript> regularity for infinity harmonic functions in two dimensions

We propose a new method for showing C ^{1, α} regularity for solutions of the infinity Laplacian equation and provide full details of the proof in two dimensions. The proof for dimensions n ≥ 3 depends upon some conjectured local gradient estimates for solutions of certain transformed PDE. LCE is supported in part by NSF Grant DMS-0500452. OS was supported in part by the Miller Institute for Basic Research in Science, Berkeley.     

Random Dirichlet problem: Scalar darcy's law

Using Ergodic Theory and Epiconvergence notion, we study the rate of convergence of solutions relative to random Dirichlet problems in domains ofR ^d with random holes whose size tends to 0. This stochastic analysis allows to extend the results already obtained in the corresponding periodic case.     

Cohomologically trivial internal categories in categories of groups with operations

We describe cohomologically trivial internal categories in the categoryC of groups with operations satisfying certain conditions ([15], [16]). As particular cases we obtain: ifC=Gr, H⁰(C, –)=0 iff C is a connected internal category; ifC=Ab,H ¹(C, –)=0 iff C is equivalent to the discrete internal category (Cokerd, Cokerd, 1, 1, 1, 1). We also discuss related questions concerning extensions, internal categories, their cohomology and equivalence in the categoryC.     

Concentration sets of the Landau–Lifshitz system and quasi-mean curvature flows

The aim of this work is to analyze the concentration set of the stationary weak solutions to the Landau-Lifshitz system of the ferromagnetic spin chain from R ^m into the unit sphere S ² of R ³ . Suppose that u _k → u weakly in W ^1,2(R ^m × R ₊, S ²) and that Σ ^t is the concentration set for fixed t. In the present paper we first prove that Σ ^t is a -rectifiable set for almost all t ∈ R ₊. And then we verify that Σ ^t moves by the quasi-mean curvature under some assumptions, which is a new codimension 2 curvature flow. Finally we analyze the behavior of the solution at the singular point and get the blow up formulas. The main barrier to Landau–Lifshtiz system is that there is no energy monotonicity inequality. After the seminal work to on the study of the concentration set of minimizing energy harmonic maps by Leon Simon, there are several papers dealing with the stationary harmonic maps and its heat flows, and so on. This investigation is inspired by the study on the heat flow of harmonic maps and it largely depends on our result of the partial regularity.     

Infinitely many solutions of dirichlet problem for <Emphasis Type="Italic">p</Emphasis>-mean curvature operator

The existence of infinitely many solutions of the following Dirichlet problem for p-mean curvature operator: is considered, where Θ is a bounded domain in R ⁿ (n>p>1) with smooth boundary ∂Θ. Under some natural conditions together with some conditions weaker than (AR) condition, we prove that the above problem has infinitely many solutions by a symmetric version of the Mountain Pass Theorem if . Supported by the National Natural Science Foundation of China (10171032) and the Guangdong Provincial Natural Science Foundation (011606).     

Fine regularity for elliptic systems with discontinuous ingredients

We propose results on interior Morrey, BMO and H?lder regularity for the strong solutions to linear elliptic systems of order 2b with discontinuous coefficients and right-hand sides belonging to the Morrey space L^p,λ. Received: 20 October 2004     

On classical solutions of the compressible Navier-Stokes equations with nonnegative initial densities

We study the Navier-Stokes equations for compressible barotropic fluids in a bounded or unbounded domain Ω of R³. We first prove the local existence of solutions (ρ,u) in C([0,T_*]; (ρ^∞ +H³(Ω)) × under the assumption that the data satisfies a natural compatibility condition. Then deriving the smoothing effect of the velocity u in t>0, we conclude that (ρ,u) is a classical solution in (0,T_**)×Ω for some T_** ∈ (0,T_*]. For these results, the initial density needs not be bounded below away from zero and may vanish in an open subset (vacuum) of Ω.     

Toeplitz operators on harmonic Bergman spaces

We study Toeplitz operators on the harmonic Bergman spaceb ^p (B), whereB is the open unit ball inR ⁿ(n2), for 1<p. We give characterizations for the Toeplitz operators with positive symbols to be bounded, compact, and in Schatten classes. We also obtain a compactness criteria for the Toeplitz operators with continuous symbols.