On Equations Which Describe Pseudospherical Surfaces

Using the notion of a differential equation which describes an η-pseudospherical surface (η-p.s.s.), we give a characterization of the equations of type u_xt = F(u, u_x,…, ?^ku / ?x^k), k ≥ 2, with this property. We obtain a systematic procedure to determine a linear problem for which a given equation is the integrability condition. The equations of type u_xt = F(u, u_x) were characterized by Rabelo and Tenenblat in another paper. The theory is applied to several equations, some of which were not known to describe η-p.s.s.     

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     

Uniform decay estimates for solutions of the Yamabe equation

We study positive solutions u of the Yamabe equation c_m Du-s( x) u+k( x) u^\fracm+2m-2=0{c_{m} \Delta u-s\left( x\right) u+k\left( x\right) u^{\frac{m+2}{m-2}}=0}, when k(x) > 0, on manifolds supporting a Sobolev inequality. In particular we get uniform decay estimates at infinity for u which depend on the behaviour at infinity of k, s and the L ^Γ-norm of u, for some G 3 \tfrac2mm-2{\Gamma\geq\tfrac{2m}{m-2}}. The required integral control, in turn, is implied by further geometric conditions. Finally we give an application to conformal immersions into the sphere.     

A characterization of convexity and central symmetry for planar polygonal sets

LetS be a compact set inR ² with nonempty interior,L(u,k) be a line 〈u, x〉 =k, and ζ _u (k) be the linear Lebesgue measure ofS∩L(u,k). It is well known that for a convexS, ζ _u (k) is unimodal, that is, as a function ofk, it is first non-decreasing and then nonincreasing for everyu∈R ². Further, ifS is centrally symmetric with respect toM, ζ _u (k) achieves maximum whenL(u, k) passes throughM. Converse propositions are considered in this paper for polygonalS with Jordan boundary. It is shown that unimodality alone does not suffice for convexity. However, if ζ _u (k) achieves maximum wheneverL(u, k) passes through some fixed pointM then unimodality yields convexity as well as central symmetry. It is also shown that continuity of ζ _u (k) in the interior of its support implies convexity ofS. This last result, however, is false for non-polygonal sets. Research supported by National Science Foundation Grant GP-28154.     

On one representation of analytic functions by harmonic functions

Let u(x) be a function analytic in some neighborhood D about the origin, $ \mathcal{D} Let u(x) be a function analytic in some neighborhood D about the origin, ⊂ ℝ ⁿ . We study the representation of this function in the form of a series u(x) = u ₀(x) + |x|² u ₁(x) + |x|⁴ u ₂(x) + …, where u _k (x) are functions harmonic in . This representation is a generalization of the well-known Almansi formula. Original Russian Text ? V. V. Karachik, 2007, published in Matematicheskie Trudy, 2007, Vol. 10, No. 2, pp. 142–162.     

A Note on an Error Bound for the AOR Method

Suppose Ax = b is a system of linear equations where the matrix A is symmetric positive definite and consistently ordered. A bound for the norm of the errors _k = x– x ^k of the AOR method in terms of the norms of _k = x ^k–x ^k–1 and _k+1 = x ^k+1–x ^k and their inner product is derived.     

An integrable equation with nonsmooth solitons

We present the bi-Hamiltonian structure and Lax pair of the equation ρ_t = bu_x+(1/2)[(u ² −u_x ² )ρ]_x, where ρ = u − u_xx and b = const, which guarantees its integrability in the Lax pair sense. We study nonsmooth soliton solutions of this equation and show that under the vanishing boundary condition u → 0 at the space and time infinities, the equation has both “W/M-shape” peaked soliton (peakon) and cusped soliton (cuspon) solutions.     

Über das homogene Dirichlet-Problem bei nichtlinearen partiellen Differentialgleichungen vom Typ der Boussinesq-Gleichung

Semilinear equations of Boussinesq type, e.g. u_tt + u_xx ? u_xxxx + (u²)_xx = 0, u_tt + u_xx ? u_xxxx + u_xu_xx = 0, or certain equations containing the squared wave operator, e.g. u_xxtt ? u^k = 0, k ? N k ≥ 2, are studied. A generalized boundary value problem on bounded domains can be treated using Hilbert space methods. The linear parts of these equations are not elliptic, the latter not even hypoelliptic. A mountain pass lemma is used to prove the existence of nontrivial weak solutions. These solutions are obtained in anisotropic Sobolev spaces.     

On uniqueness problems related to the Fokker–Planck–Kolmogorov equation for measures

We survey recent results related to uniqueness problems for parabolic equations for measures. We consider equations of the form ∂ _t μ = L ^* μ for bounded Borel measures on ℝ ^d  × (0, T), where L is a second order elliptic operator, for example, Lu = D_xu + ( b,?_xu ) Lu = {\Delta_x}u + \left( {b,{\nabla_x}u} \right) , and the equation is understood as the identity

Isoperimetric inequalities, Wulff shape and related questions for strongly nonlinear elliptic operators

We investigate the first eigenvalue of a highly nonlinear class of elliptic operators which includes the p--Laplace operator D_p u=?_i [(?)/(?x_i)] (|?u |^p-2[(?u)/(?x_i)])\Delta_p u=\sum_i {{\partial}\over{\partial x_i}} (\vert\nabla u \vert^{p-2}{{\partial u}\over{\partial x_i}}), the pseudo-p-Laplace operator [(D)\tilde]_p u=?_i [(?)/(?x_i)] (|[(?u)/(?x_i)] |^p-2 [(?u)/(?x_i)])\tilde\Delta_p u=\sum_i {{\partial}\over{\partial x_i}} (\vert {{\partial u}\over{\partial x_i}} \vert^{p-2} {{\partial u}\over{\partial x_i}}) and others. We derive the positivity of the first eingefunction, simlicity of the first eigenvalue, Faber-Krahn and Payne-Rayner type inequalities. In another chapter we address the question of symmetry for positive solutions to more general equations. Using a Pohozaev-type inequality and isoperimetric inequalities as well as convex rearrangement methods we generalize a symmetry result of Kesavan and Pacella. Our optimal domains are level sets of a convex function H ^o. They have the so-called Wulff shape associated with H and only in special cases they are Euclidean balls.     

Bäcklund Transformations and Inverse Scattering Solutions for Some Pseudospherical Surface Equations

In is known that the equations [u_t ? g(u)u_x]x = ±g′(u) describe pseudo-spherical surfaces, i.e. that these equations are the integrability conditions for the structural equations of such surfaces, provided g satisfies g′ + µg = θ. In this paper we obtain self-Bäcklund transformations for these equations by a geometric method, and show how the inverse scattering method generates global solutions.     

An inverse polynomial method for the identification of the leading coefficient in the Sturm–Liouville operator from boundary measurements

An inverse polynomial method of determining the unknown leading coefficient k=k(x) of the linear Sturm–Liouville operator Au=−(k(x)u^′(x))^′+q(x)u(x), x(0,1), is presented. As an additional condition only two measured data at the boundary (x=0,x=1) are used. In absence of a singular point (u^′(x)≠0,u^″(x)≠0,x[0,1]) the inverse problem is classified as a well-conditioned . If there exists at least one singular point, then the inverse problem is classified as moderately ill-conditioned (u^′(x₀)=0,x₀(0,1);u^′(x)≠0,x≠x₀;u^″(x)≠0,x[0,1]) and severely ill-conditioned (u^′(x₀)=u^″(x₀)=0,x₀(0,1);u^′(x)≠0,u^″(x)≠0,x≠x₀). For each of the cases direct problem solution is approximated by corresponding polynomials and the inverse problem is reformulated as a Cauchy problem for to the first order differential equation with respect the unknown function k=k(x). An approximate analytical solution of the each Cauchy problems are derived in explicit form. Numerical simulations all the above cases are given for noise free and noisy data. An accuracy of the presented approach is demonstrated on numerical test solutions.     

An approximation result for solutions of Hessian equations

We show that W ^2,p weak solutions of the k-Hessian equation F _k (D ² u) = g(x) with k≥ 2 can be approximated by smooth k-convex solutions v _j of similar equations with the right hands sides controlled uniformly in C ^0,1 norm, and so that the quantities are bounded independently of j. This result simplifies the proof of previous interior regularity results for solutions of such equations. It also permits us to extend certain estimates for smooth solutions of degenerate two dimensional Monge–Ampère equations to W ^2,p solutions. Supported by an Australian Research Council Senior Fellowship.     

Semigroup approach for identification of the unknown diffusion coefficient in a quasi‐linear parabolic equation

This article presents a semigroup approach for the mathematical analysis of the inverse coefficient problems of identifying the unknown coefficient k(u(x,t)) in the quasi‐linear parabolic equation u_t(x,t)=(k(u(x,t))u_x(x,t))_x, with Dirichlet boundary conditions u(0,t)=ψ₀, u(1,t)=ψ₁. The main purpose of this paper is to investigate the distinguishability of the input–output mappings Φ[?]:?? →C¹[0,T], Ψ[?]:??→C¹[0,T] via semigroup theory. In this paper, it is shown that if the null space of the semigroup T(t) consists of only zero function, then the input–output mappings Φ[?] and Ψ[?] have the distinguishability property. It is also shown that the types of the boundary conditions and the region on which the problem is defined play an important role in the distinguishability property of these mappings. Moreover, under the light of measured output data (boundary observations) f(t):=k(u(0,t))u_x(0,t) or/and h(t):=k(u(1,t))u_x(1,t), the values k(ψ₀) and k(ψ₁) of the unknown diffusion coefficient k(u(x,t)) at (x,t)=(0,0) and (x,t)=(1,0), respectively, can be determined explicitly. In addition to these, the values k_u(ψ₀) and k_u(ψ₁) of the unknown coefficient k(u(x,t)) at (x,t)=(0,0) and (x,t)=(1,0), respectively, are also determined via the input data. Furthermore, it is shown that measured output data f(t) and h(t) can be determined analytically by an integral representation. Hence the input–output mappings Φ[?]:??→ C¹[0,T], Ψ[?]:??→C¹[0,T] are given explicitly in terms of the semigroup. Copyright © 2007 John Wiley & Sons, Ltd.     

Best constants in Korn-Poincaré's inequalities on a slab

We establish the best constants in the Poincaré-type and the trace-type inequalities for the quadratic form $ \lambda ||\,{\rm div}\,{\rm u}\,||_{L^2 }^2 \, + \,2\,\mu \,||\,\,(\nabla {\rm u}\, + \,\nabla {\rm u}^{\rm T})/2\,||_{L^2 }^2 $ which is fundamental in elasticity theory, on the space of H¹ vector fields u on a slab vanishing on one or both of its sides. We similarly calculate those constants for the case of H¹ divergence-free vector fields. Our method, which is fairly general, has another practical application to the quadratic form ∑_j,k(a^jk?_ku, ?_ju)_L² with coefficients a ^jk = a^kj ε C in H¹ scalar functions u on a slab.     

Removable singularities of weak solutions to the navier-stokes equations

Consider the Navier-Stokes equations in Ω×(0,T), where Ω is a domain in R³. We show that there is an absolute constant ε₀ such that ever, y weak solution u with the property that Sup_tε(a,b)|u(t)|_L(D)≤ε0 is necessarily of class C^∞ in the space-time variables on any compact suhset of D × (a,b) , where D?? and 0 a<b<T. As an application. we prove that if the weak solution u behaves around (x_o, t_o) εΩ×(o,T) 1ike u(x, t) = o(|x - xo|^-1) as x→x ₀ uniforlnly in t in some neighbourliood of t_o, then (x_o,t_o) is actually a removable singularity of u.     

Reaction diffusion equations with non-linear boundary conditions,blowup and steady states

In this paper we study the following problem: u_t−Δu=−f(u) in Ω×(0, T)≡Q_T, ∂u ∂n=g(u) on ∂Ω×(0, T)≡S_T, u(x, 0)=u₀(x) in Ω , where Ω⊂ℝ^N is a smooth bounded domain, f and g are smooth functions which are positive when the argument is positive, and u₀(x)>0 satisfies some smooth and compatibility conditions to guarantee the classical solution u(x, t) exists. We first obtain some existence and non-existence results for the corresponding elliptic problems. Then, we establish certain conditions for a finite time blow-up and global boundedness of the solutions of the time-dependent problem. Further, we analyse systems with same kind of boundary conditions and find some blow-up results. In the last section, we study the corresponding elliptic problems in one-dimensional domain. Our main method is the comparison principle and the construction of special forms of upper–lower solutions using related equations. © 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.     

Fourier Analysis of Semistable Distributions

We consider the generalized convolution powers G _α ^*u (x) of an arbitrary semistable distribution function G _α (x) of exponent α∈(0,2), and prove that for all j, k∈{0,1,2,…} and u>0 the derivatives G _α ^(k,j)(x;u)=∂ ^k+j G _α ^*u (x)/∂ x ^k ∂ u ^j , x∈ℝ, are of bounded variation on the whole real line ℝ. The proof, along with an integral recursion in j, is new even in the special case of stable laws, and the result provides a framework for possible asymptotic expansions in merge theorems from the domain of geometric partial attraction of semistable laws. An erratum to this article can be found at     

Finite‐difference approximation for the u(k)‐derivative with O(hM−k+1) accuracy: An analytical expression

An approximation of function u(x) as a Taylor series expansion about a point x₀ at M points x_i, ～ i = 1,2,…,M is used where x_i are arbitrary‐spaced. This approximation is a linear system for the derivatives u^(k) with an arbitrary accuracy. An analytical expression for the inverse matrix A ^?1 where A = [A_ik] = (x_i ? x₀)^k is found. A finite‐difference approximation of derivatives u^(k) of a given function u(x) at point x₀ is derived in terms of the values u(x_i). © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006