Generalized Solutions of Nonlocal Elliptic Problems

An elliptic equation of order 2m with general nonlocal boundary-value conditions, in a plane bounded domain G with piecewise smooth boundary, is considered. Generalized solutions belonging to the Sobolev space W ₂ ^m (G) are studied. The Fredholm property of the unbounded operator (corresponding to the elliptic equation) acting on L ₂(G), and defined for functions from the space W ₂ ^m (G) that satisfy homogeneous nonlocal conditions, is established.__________Translated from Matematicheskie Zametki, vol. 77, no. 5, 2005, pp. 665–682.Original Russian Text Copyright ©2005 by P. L. Gurevich.     

On the Well-Posedness of a Two-Point Boundary-Value Problem for a System with Pseudodifferential Operators

We investigate the problem of the well-posedness of a boundary-value problem for a system of pseudodifferential equations of arbitrary order with nonlocal conditions. The equation and boundary conditions contain pseudodifferential operators whose symbols are defined and continuous in a certain domain H ⊂ ℝ _σ ^m . A criterion for the existence and uniqueness of solutions and for the continuous dependence of the solution on the boundary function is established. __________ Published in Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 8, pp. 1131 – 1136, August, 2005.     

Solvability of inhomogeneous boundary-value problems for fourth-order differential equations

We consider a Cauchy-type boundary-value problem, a problem with three boundary conditions, and the Dirichlet problem for a general typeless fourth-order differential equation with constant complex coefficients and nonzero right-hand side in a bounded domain Ω ⊂ R ² with smooth boundary. By the method of the Green formula, the theory of extensions of differential operators, and the theory of L-traces (i.e., traces associated with the differential operation L), we establish necessary and sufficient (for elliptic operators) conditions of the solvability of each of these problems in the space H ^m (Ω), m ≥ 4.     

Regular boundary value problems for complete second order elliptic differential-operator equations in UMD Banach spaces

We consider coerciveness and Fredholmness of nonlocal boundary value problems for complete second order elliptic differential-operator equations in UMD Banach spaces. In some special cases, the main coefficients of the boundary conditions may be bounded operators and not only complex numbers. Then, we prove an isomorphism, in particular, maximal L _p -regularity, of the problem with a linear parameter in the equation. In both cases, the boundary conditions may also contain unbounded operators in perturbation terms. Finally, application to regular nonlocal boundary value problems for elliptic equations of the second order in non-smooth domains are presented. Equations and boundary conditions may contain differential-integral parts. The spaces of solvability are Sobolev type spaces W _p,q ^2,2. The first author is a member of G.N.A.M.P.A. and the paper fits the 60% research program of G.N.A.M.P.A.-I.N.D.A.M.; The third author was supported by the Israel Ministry of Absorption.     

An elliptic continuation result for harmonic functions in two dimensions

Let u be harmonic in a simply connected domainG ⊂ ℝ² and letK be a compact subset of G. In this note, it is proved there exists an “elliptic continuation” of u, namely there exist a smooth functionu ₁ and a second order uniformly elliptic operatorL with smooth coefficients in ℝ², satisfying:u ₁=u inK, Lu ₁=0 in ℝ². A similar continuation theorem, with u itself a solution to an elliptic second order equation inG, is also proved.     

On the existence of a Feller semigroup with atomic measure in a nonlocal boundary condition

The existence of Feller semigroups arising in the theory of multidimensional diffusion processes is studied. An elliptic operator of second order is considered on a plane bounded region G. Its domain of definition consists of continuous functions satisfying a nonlocal condition on the boundary of the region. In general, the nonlocal term is an integral of a function over the closure of the region G with respect to a nonnegative Borel measure μ(y, dη) ∈ ∂G. It is proved that the operator is a generator of a Feller semigroup in the case where the measure is atomic. The smallness of the measure is not assumed. Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2008, Vol. 260, pp. 164–179.     

Asymptotic analysis of a parabolic semilinear problem with nonlinear boundary multiphase interactions in a perforated domain

We consider a parabolic semilinear problem with rapidly oscillating coefficients in a domain Ω_ε that is ε-periodically perforated by small holes of size O\mathcal {O}(ε). The holes are divided into two ε-periodical sets depending on the boundary interaction at their surfaces, and two different nonlinear Robin boundary conditions σ_ε(u _ε) + εκ _m (u _ε) = εg ^(m) _ε, m = 1, 2, are imposed on the boundaries of holes. We study the asymptotics as ε → 0 and establish a convergence theorem without using extension operators. An asymptotic approximation of the solution and the corresponding error estimate are also obtained. Bibliography: 60 titles. Illustrations: 1 figure.     

Asymptotic analysis of a boundary-value problem with nonlinear multiphase boundary interactions in a perforated domain

We consider a boundary-value problem for the second-order elliptic differential operator with rapidly oscillating coefficients in a domain Ω _ε that is ε-periodically perforated by small holes. The holes are split into two ε-periodic sets depending on the boundary interaction via their boundary surfaces. Therefore, two different nonlinear boundary conditions σ _ε (u _ε ) + εκ _m (u _ε ) = εg _ε ^(m) , m = 1, 2, are given on the corresponding boundaries of the small holes. The asymptotic analysis of this problem is performed as ε → 0, namely, the convergence theorem for both the solution and the energy integral is proved without using an extension operator, asymptotic approximations for the solution and the energy integral are constructed, and the corresponding approximation error estimates are obtained.     

On coerciveness and semiboundedness of general boundary problems

The paper treats coerciveness inequalities (of the form Re(Au, u)≧c ‖u‖ _s ² −λ ‖u‖ ₀ ² ,c>0,λ ∈ R) and semiboundedness inequalities (of the form Re (Au, u)≧−λ ‖u‖²) for the general boundary problems associated with an elliptic 2m-order differential operatorA in a compactn-dimensional manifold with boundary. In particular, we study the normal pseudo-differential boundary conditions, for which we determine necessary and sufficient conditions for coerciveness withs=m, and for semiboundedness with ‖u‖ = ‖u‖_m, in explicit form.     

Symmetric products of surfaces and the cycle index

We study some of the combinatorial structures related to the signature ofG-symmetric products of (open) surfacesSP _G ^m (M)=M ^m/G whereG ⊂S _m.The attention is focused on the question, what information about a surfaceM can be recovered from a symmetric productSP ⁿ(M). The problem is motivated in part by the study of locally Euclidean topological commutative (m+k,m)-groups, [16]. Emphasizing a combinatorial point of view we express the signature Sign(SP _G ^m (M))in terms of the cycle index ofG, a polynomial which originally appeared in Pólya enumeration theory of graphs, trees, chemical structures etc. The computations are used to show that there exist punctured Riemann surfacesM _g,k,M _g′,k′such that the manifoldsSP ^m(M _g,k)andSP ^m(M)_g′,k′)are often not homeomorphic, although they always have the same homotopy type provided 2 _g +k=2 _g′ +k′ andk,k′≥1. Supported by the Serbian Ministry for Science and Technology, Grant No. 1643.     

Dissipative Dynamics for a Class of Nonlinear Pseudo-Differential Equations

We study a class of nonlinear evolutionary equations generated by an elliptic pseudo-differential operator, and with nonlinearity of the form G(u _x ) where cη² ≤ G(η) ≤ Cη² for large |η|. For the evolution in spaces of periodic functions with zero mean we demonstrate existence of a universal absorbing set and compact attractor. Furthermore, we show that the attractor is of a finite Hausdorf dimension. The dissipation mechanism for the class of equations studied in the paper is akin to the nonlinear saturation in the Kuramoto-Sivashinsky equation. A similar generalization of the Kuramoto-Sivashinsky equation was studied by Nicolaenko et al. under the assumption of a purely quadratic nonlinearity and reflection invariance of both: the equation and solutions.      

Square roots of elliptic second order divergence operators on strongly lipschitz domains:<Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript> theory

We prove the Kato conjecture for square roots of elliptic second order non-self-adjoint operators in divergence formL = -div(A∇) on strongly Lipschitz domains in ℝⁿ, n≥2, subject to Dirichlet or to Neumann boundary conditions. The method relies on a transference procedure from the recent positive result on ℝⁿ in [2].     

Solvability of nonlocal elliptic problems in weight spaces

In this paper we consider elliptic equations of order 2m in a bounded domainQ є R ⁿ with boundaryδQ and nonlocal conditions relating the traces of the solution and its derivatives on (n − 1)-dimensional smooth manifolds Γ _i (∪ _i =∂δQ) to their values on some compact setF ⊂Q, whereF ∩δQ ≠ Φ. The Fredholm solvability of these problems in the weight spacesV _{p, a} ^/l+2m (Q) is proved for arbitrary 1<p <∞. Translated fromMatematicheskie Zametki, Vol. 67, No. 6, pp. 882–898, June, 2000. This research was supported by the Russian Foundation for Basic Research under grant No. 99-01-00028.     

The Upper Bounds of Arbitrary Eigenvalues for Uniformly Elliptic Operators with Higher Orders

Let Ωbelong to R^m （m≥ 2） be a bounded domain with piecewise smooth and Lipschitz boundary δΩ Let t and r be two nonnegative integers with t ≥ r ＋ 1. In this paper, we consider the variable-coefficient eigenvalue problems with uniformly elliptic differential operators on the left-hand side and （-Δ）^T on the right-hand side. Some upper bounds of the arbitrary eigenvalue are obtained, and several known results are generalized.     

A Hilbert Transform for Matrix Functions on Fractal Domains

We consider H?lder continuous circulant (2 × 2) matrix functions G¹₂{{\bf G}^1_2} defined on the fractal boundary Γ of a Jordan domain Ω in \mathbbR²ⁿ{\mathbb{R}^{2n}}. The main goal is to establish a Hilbert transform for such functions, within the framework of Hermitian Clifford analysis. This is a higher dimensional function theory centered around the simultaneous null solutions of two first order vector valued differential operators, called Hermitian Dirac operators. In Brackx et al. (Bull Braz Math Soc 40(3): 395–416, 2009) a Hermitian Cauchy integral was constructed by means of a matrix approach using circulant (2 × 2) matrix functions, from which a Hilbert transform was derived in Brackx et al. (J Math Anal Appl 344: 1068–1078, 2008) for the case of domains with smooth boundary. However, crucial parts of the method are not extendable to the case where the boundary of the considered domain is fractal. At present we propose an alternative approach which will enable us to define a new Hermitian Hilbert transform in that case. As a consequence, we give necessary and sufficient conditions for the Hermitian monogenicity of a circulant matrix function G¹₂{{\bf G}^1_2} in the interior and exterior of Ω, in terms of its boundary value g¹₂=G¹₂|_G{{\bf g}^1_2={\bf G}^1_2|_\Gamma}, extending in this way also results of Abreu Blaya et al. (Bound. Value Probl. 2008: 2008) (article ID 425256), (article ID 385874), where Γ is required to be Ahlfors–David regular.     

Large deviations and continuum limit in the 2D Ising model

Summary. We study the 2D Ising model in a rectangular box Λ _L of linear size O(L). We determine the exact asymptotic behaviour of the large deviations of the magnetization ∑ _t∈ΛL σ(t) when L→∞ for values of the parameters of the model corresponding to the phase coexistence region, where the order parameter m ^* is strictly positive. We study in particular boundary effects due to an arbitrary real-valued boundary magnetic field. Using the self-duality of the model a large part of the analysis consists in deriving properties of the covariance function <σ(0)σ(t)>, as |t|→∞, at dual values of the parameters of the model. To do this analysis we establish new results about the high-temperature representation of the model. These results are valid for dimensions D≥2 and up to the critical temperature. They give a complete non-perturbative exposition of the high-temperature representation. We then study the Gibbs measure conditioned by {|∑ _t∈ΛL σ(t) −m|Λ _L ||≤|Λ _L |L ^{− c} }, with 0<c<1/4 and −m ^*<m<m ^*. We construct the continuum limit of the model and describe the limit by the solutions of a variational problem of isoperimetric type. Received: 17 October 1996 / In revised form: 7 March 1997     

An inhomogeneous elliptic complex

We define a 3 term sequenceP of differential operators of mixed type; the first and third operators are 1st order while the second operator is 2nd order.P is always elliptic; it forms a complex ifM is einstein. It was first discussed by Gasqui.P is related to similar complexesC andG discussed by 02 Calabi and Gasqui-Goldschmidt. The index and equivariant index ofP vanish. In dimension 2,P=C⊗s whereS is of Dirac type;C and-S determine the same equivariant index. We study the heat equation asymptotics of the operators ofP; the associated Laplacians do not have scalar leading symbol. Research partially supported by the NSF, NSA, IHES, and Ohio State     

Some questions in the theory of inverse problems for differential operators

For the two operatorsLy=y ⁿ +σ _k=0 ⁿ⁻² p _k (x)y( ^k ) and Ry=yⁿ+σ _k=0 ⁿ⁻² p_k(x)y(^k) with a common set of boundary conditions we establish a connection between p_k(x) and P_k(x) in the case where the weight numbers coincide and a finite number of the eigenvalues do not coincide, in terms of the eigenfunctions of these operators corresponding to the noncoincident eigenvalues and the derivatives of these functions. This enables us to recover the operator L from the operator R by solving a system of nonlinear ordinary differential equations. For Sturm-Liouville operators an analogous relation is proved for the case where infinitely many eigenvalues do not coincide. Translated from Matematicheskie Zametki, Vol. 21, No. 2, pp. 151–160, February, 1977. I wish to express my thanks to my scientific adviser V. A. Sadovnich.     

Interior a priori estimates in discreteL p norms for solutions of parabolic and elliptic difference equations

Summary We consider elliptic and parabolic difference operators and prove estimates in discrete L_p norms, 1<p<∞, which are analogues of known estimates for the corresponding differential operators. Let U be a solution in a bounded domain Ω of an elliptic or parabolic differential equation and let U_h be a solution of the discrete equation. Using the estimates, we prove under mild regularity assumptions that if U_h converges to U in some discrete L_p normp>1, then the difference quotients of U_h converge uniformly (on compact subsets of Ω) to the corresponding derivatives of U. Entrata in Redazione il 9 ottobre 1971.