Solution of two-dimensional boundary value problems corresponding to initial-boundary value problems of diffusion on a right cylinder

We consider boundary value problems for the differential equations Δ₂ u + B u = 0 with operator coefficients B corresponding to initial-boundary value problems for the diffusion equation Δ₃ u − pu = ∂ _t u (p > 0) on a right cylinder with inhomogeneous boundary conditions on the lateral surface of the cylinder with zero boundary conditions on the bases of the cylinder and with zero initial condition. For their solution, we derive specific boundary integral equations in which the space integration is performed only over the lateral surface of the cylinder and the kernels are expressed via the fundamental solution of the two-dimensional heat equation and the Green function of corresponding one-dimensional initial-boundary value problems of diffusion. We prove uniqueness theorems and obtain sufficient existence conditions for such solutions in the class of functions with continuous L ₂-norm.     

A new technique of integral representations in ?n

A new technique of integral representations in ℂ ⁿ , which is different from the well-known Henkin technique, is given. By means of this new technique, a new integral formula for smooth functions and a new integral representation of solutions of the ∂-equations on strictly pseudoconvex domains in ℂ ⁿ are obtained. These new formulas are simpler than the classical ones, especially the solutions of the ∂-equations admit simple uniform estimates. Moreover, this new technique can be further applied to arbitrary bounded domains in ℂ ⁿ so that all corresponding formulas are simplified.     

On the Unique Continuation Properties for Elliptic Operators with Singular Potentials

Let u be a solution to a second order elliptic equation with singular potentials belonging to Kato-Fefferman-Phong＇s class in Lipschitz domains. An elementary proof of the doubling property for u^2 over balls is presented, if the balls are contained in the domain or centered at some points near an open subset of the boundary on which the solution u vanishes continuously. Moreover, we prove the inner unique continuation theorems and the boundary unique continuation theorems for the elliptic equations, and we derive the Bp weight properties for the solution u near the boundary.     

The Cauchy Boundary Value Problems on Closed Piecewise Smooth Manifolds in <Emphasis Type="Italic">C</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

Suppose that D is a bounded domain with a piecewise C^1 smooth boundary in C^n. Let ψ∈C^1 α(δD). By using the Hadamard principal value of the higher order singular integral and solid angle coefficient method of points on the boundary, we give the Plemelj formula of the higher order singular integral with the Boehner-Martinelli kernel, which has integral density ψ. Moreover, by means of the Plemelj formula and methods of complex partial differential equations, we discuss the corresponding Cauehy boundary value problem with the Boehner-Martinelli kernel on a closed piecewise smooth manifold and obtain its unique branch complex harmonic solution.     

Integral representations of harmonic functions in half spaces

In this paper, using a modified Poisson kernel in an upper half-space, we prove that a harmonic function u(z) in a upper half space with its positive part u⁺(x)=max{u(x),0} satisfying a slowly growing condition can be represented by its integral in the boundary of the upper half space, the integral representation is unique up to the addition of a harmonic polynomial, vanishing in the boundary of the upper half space and that its negative part u⁻(x)=max{−u(x),0} can be dominated by a similar slowly growing condition, this improves some classical result about harmonic functions in the upper half space.     

Regularity in a Singular Biharmonic Dirichlet Problem

We consider the singular biharmonic equation with Dirichlet boundary conditions u = f₀ and ∂_nu = f₁ on . In our setup the boundary values f_j (j = 0,1) are elements in two homogeneous Banach spaces B_j (j = 0,1) on . We give a sufficient condition on the spaces B_j (j = 0,1) to ensure that the solution u of this Dirichlet problem has the appropriate boundary values f_j (j = 0,1) in the sense of convergence in spaces B_j (j = 0,1). Our results also apply in the unweighted case.     

A new look at smoothing Newton methods for nonlinear complementarity problems and box constrained variational inequalities

In this paper we take a new look at smoothing Newton methods for solving the nonlinear complementarity problem (NCP) and the box constrained variational inequalities (BVI). Instead of using an infinite sequence of smoothing approximation functions, we use a single smoothing approximation function and Robinson’s normal equation to reformulate NCP and BVI as an equivalent nonsmooth equation H(u,x)=0, where H:ℜ ²ⁿ →ℜ ²ⁿ , u∈ℜ ⁿ is a parameter variable and x∈ℜ ⁿ is the original variable. The central idea of our smoothing Newton methods is that we construct a sequence {z ^k =(u ^k ,x ^k )} such that the mapping H(·) is continuously differentiable at each z ^k and may be non-differentiable at the limiting point of {z ^k }. We prove that three most often used Gabriel-Moré smoothing functions can generate strongly semismooth functions, which play a fundamental role in establishing superlinear and quadratic convergence of our new smoothing Newton methods. We do not require any function value of F or its derivative value outside the feasible region while at each step we only solve a linear system of equations and if we choose a certain smoothing function only a reduced form needs to be solved. Preliminary numerical results show that the proposed methods for particularly chosen smoothing functions are very promising. Received June 23, 1997 / Revised version received July 29, 1999?Published online December 15, 1999     

Three-term asymptotics for the boundary layers of semilinear elliptic eigenvalue problems

We consider the nonlinear eigenvalue problem −Δu=λ f(u) in Ω u=0 on ∂Ω, where Ω is a ball or an annulus in R^N (N ≥ 2) and λ > 0 is a parameter. It is known that if λ >> 1, then the corresponding positive solution u_λ develops boundary layers under some conditions on f. We establish the asymptotic formulas for the slope of the boundary layers of u_λ with the exact second term and the ‘optimal’ estimate of the third term.     

Radial solutions to the wave equation

We study some boundedness properties of radial solutions to the Cauchy problem associated to the wave equation (∂ _t ²-▵ _x )u(t,x)=0 and meanwhile we give a new proof of the solution formula. Received: July 7, 1998?Published online: March 19, 2002     

Supernumary polylogarithmic ladders and related functional equations

Summary The nature of the polylogarithmic ladder is briefly reviewed, and its close relationship to the associated cyclotomic equation explained. Generic results for the base determined by the family of equationsu ^p +u ^q = 1 are developed, and many new supernumary ladders, existing for particular values ofp andq, are discussed in relation to theirad hoc cyclotomic equations. Results for ordersn from 6 through 9, for which no relevant functional equations are known, are reviewed; and new results for the base , where ³ + = 1, are developed through the sixth order.Special results for the exponentp from 4 through 6 are determined whenever a new cyclotomic equation can be constructed. Only the equationu ⁵+u ³ = 1 has so far resisted this process. The need for the constraint (p,q) = 1 is briefly considered if redundant formulas are to be avoided.The equationu ^6m+1 +u ^6r–1 = 1 is discussed and some valid results deduced. This equation is divisible byu ² –u + 1, and the quotient polynomial is useful for constructing cyclotomic equations. The casem = 1,r = 2 is the first example encountered for which no valid ladders have yet been found.New functional equations to give the supernumary -ladders of index 24 are developed, but their construction runs into difficulty at the third order, apparently requiring the introduction of an adjoint set of variables that blocks the extension to the fourth order.A demonstration, based on the indices of existing accessible and supernumary ladders, indicates that functional equations based on arguments ±z ^m (1–z) ^r (1 +z) ^s are not capable of extension to the sixth order.There are some miscellaneous supernumary ladders that seem incapable, at this time, of analytic proof, and these are briefly discussed. In conclusion, applications of ladders are considered, and attention drawn to the existence of ladders with the base on the unit circle giving rise to Clausenfunction formulas which may play an important role inK-theory.     

The asymptotic behavior of Chern-Simons Higgs model on a compact Riemann surface with boundary

We study the self-dual Chern-Simons Higgs equation on a compact Riemann surface with the Neumann boundary condition.In the previous paper,we show that the Chern-Simons Higgs equation with parameter λ0 has at least two solutions(uλ1,uλ2) for λ sufficiently large,which satisfy that uλ1→u0 almost everywhere as λ→∞,and that uλ2→∞ almost everywhere as λ→∞,where u 0 is a(negative) Green function on M.In this paper,we study the asymptotic behavior of the solutions as λ→∞,and prove that uλ2-uλ2 converges to a solution of the Kazdan-Warner equation if the geodesic curvature of the boundary M is negative,or the geodesic curvature is nonpositive and the Gauss curvature is negative where the geodesic curvature is zero.     

Differential operators associated with zonal polynomials. II

Summary LetC _κ(S) be the zonal polynomial of the symmetricm×m matrixS=(s_ij), corresponding to the partition κ of the non-negative integerk. If ∂/∂S is them×m matrix of differential operators with (i, j)th entry ((1+δ_ij)∂/∂s_ij)/2, δ being Kronecker's delta, we show that C_k(∂/∂S)C_λ(S)=k!δ_λkC_k(I), where λ is a partition ofk. This is used to obtain new orthogonality relations for the zonal polynomials, and to derive expressions for the coefficients in the zonal polynomial expansion of homogenous symmetric polynomials.     

A priori and universal estimates for global solutions of superlinear degenerate parabolic equations

We prove an a priori estimate and a universal bound for any global solution of the nonlinear degenerate reaction-diffusion equation u _t =Δu ^m +u ^p in a bounded domain with zero Dirichlet boundary conditions. Received: October 1, 2001?Published online: July 9, 2002     

On Some Second Order Differential Equations with Convergent Solutions

Via a special integral transformation, asymptotic integration results for ordinary differential equations are used to establish accurate asymptotic developments for radial solutions of the elliptic equation Δu + K(|x|)e ^u = 0, |x| > x ₀ > 0, in the bidimensional case.     

Solution to the boundary blowup problem for k-curvature equation

We consider the boundary blowup problem for k-curvature equation, i.e., H _k [u] = f(u) g(|Du|) in an n-dimensional domain Ω, with the boundary condition u(x) → ∞ as dist (x,∂Ω) → 0. We prove the existence result under some hypotheses. We also establish the asymptotic behavior of a solution near the boundary ∂Ω. Mathematics Subject Classification (2000) 35J65, 35B40, 53C21     

Concentration at curves for a singularly perturbed Neumann problem in three-dimensional domains

We prove new concentration phenomena for the equation −ɛ² Δu + u = u^p in a smooth bounded domain and with Neumann boundary conditions. Here p > 1 and ɛ > 0 is small. We show that concentration of solutions occurs at some geodesics of ∂Ω when ɛ → 0. Received: September 2004 Accepted: March 2005     

Precise asymptotic formulas for variational eigencurves of semilinear two-parameter elliptic eigenvalue problems

We consider the two-parameter nonlinear eigenvalue problem?−Δu = μu − λ(u + u ^p + f(u)), u > 0 in Ω, u = 0 on ∂Ω,?where p>1 is a constant and μ,λ>0 are parameters. We establish the asymptotic formulas for the variational eigencurves λ=λ(μ,α) as μ→∞, where α>0 is a normalizing parameter. We emphasize that the critical case from a viewpoint of the two-term asymptotics of the eigencurve is p=3. Moreover, it is shown that p=5/3 is also a critical exponent from a view point of the three-term asymptotics when Ω is a ball or an annulus. This sort of criticality for two-parameter problems seems to be new. Received: February 9, 2002; in final form: April 3, 2002?Published online: April 14, 2003     

A variational approach in weighted Sobolev spaces to the operator −Δ+∂/∂x 1 in exterior domains of ℝ3

Consider the Dirichlet problem −vΔu+k∂ ₁ u = f withv, k>0 in ℝ³ or in an exterior domain of ℝ³ where the skew-symmetric differential operator −₁=∂/∂x₁ is a singular perturbation of the Laplacian. Because of the inhomogeneity of the fundamental solution we study existence, uniqueness and regularity in Sobolev spaces with anisotropic weights. In these spaces the operator ∂₁ yields an additional positive definite term giving better results than in Sobolev spaces with radial weights. The elliptic equation −vΔu +k∂₁ u=f can be taken as a model problem for the Oseen equations, a linearized form of the Navier-Stokes equations. Supported by the Sonderforschungsbereich 256 of the Deutsche Forschungsgemeinschaft at the University of Bonn     

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.     

Nonnegative solutions to an elliptic problem with nonlinear absorption and a nonlinear incoming flux on the boundary

In this paper we perform an extensive study of the existence, uniqueness (or multiplicity) and stability of nonnegative solutions to the semilinear elliptic equation − Δu = λ u − u ^p in Ω, with the nonlinear boundary condition ∂u/∂ν = u ^r on ∂Ω. Here Ω is a smooth bounded domain of with outward unit normal ν, λ is a real parameter and p, r > 0. We also give the precise behavior of solutions for large |λ| in the cases where they exist. The proofs are mainly based on bifurcation techniques, sub-supersolutions and variational methods.