Theorems Of Phragmen-Lindelof Type Research Articles

Pseudodifference Operators on Weighted Spaces, and Applications to Discrete Schrödinger Operators

We study pseudodifference operators on ZN with symbols which are bounded on ZN×TN together with their derivatives with respect to the second variable. In the same way as partial differential operators on RN are included in an algebra of pseudodifferential operators, difference operators on ZN are included in an algebra of pseudodifference operators. Particular attention is paid to the Fredholm properties of pseudodifference operators on general exponentially weighted spaces lwp(ZN) and to Phragmen–Lindelof type theorems on the exponential decay at infinity of solutions to pseudodifference equations. The results are applied to describe the essential spectrum of discrete Schrodinger operators and the decay of their eigenfunctions at infinity.

Estimates for the Energy Integral of Quasiregular Mappings on Riemannian Manifolds and Isoperimetry

The rate of growth of the energy integral of a quasiregular mapping \(f:\mathcal{X} \to \mathcal{Y}\) is estimated in terms of a special isoperimetric condition on \(\mathcal{Y}\). The estimate leads to new Phragmen-Lindelof type theorems.

Capacitary estimates for solutions of the Dirichlet problem for second order elliptic equations in divergence form

We consider the Dirichlet problem for A-harmonic functions, i.e. the solutions of the uniformly elliptic equation div(A(x)del u(x)) = 0 in an n-dimensional domain Omega, n greater than or equal to 3. The matrix A is assumed to have bounded measurable entries. We obtain pointwise estimates for the A-harmonic functions near a boundary point. The estimates are in terms of the Wiener capacity and the so called capacitary interior diameter. They imply pointwise estimates for the A-harmonic measure of the domain Omega, which in turn lead to a sufficient condition for the Holder continuity of A-harmonic functions at a boundary point. The behaviour of A-harmonic functions at infinity and near a singular point is also studied and theorems of Phragmen-Lindelof type, in which the geometry of the boundary is taken into account, are proved. We also obtain pointwise estimates for the Green function for the operator -div(A(.)del u(.)) in a domain Omega and for the solutions of the nonhomogeneous equation -div(A(x)del u(x)) = mu with measure on the right-hand side.

On the Phragmen-Lindelof principle for subharmonic functions

We consider subharmonic functions in a domain such that does not exceed some constant at all points of , . Theorems of Phragmen-Lindelof type provide an upper bound (depending on the structure of the domain ) for the a priori possible growth of as such that functions satisfying this estimate do not exceed in the whole domain . We obtain a Phragmen-Lindelof type theorem in which the restriction on the possible growth of as is expressed in terms of the lower density (with respect to plane Lebesgue measure) of the set at the point .

Spatial decay estimates for a coupled system of second-order quasilinear partial differential equations arising in thermoelastic finite anti-plane shear

The spatial decay behavior of solutions of a coupled system of second-order quasilinear partial differential equations, in divergence form, defined on a two-dimensional semi-infinite strip, is investigated. Such equations arise in the theory of anti-plane shear deformations for isotropic nonlinearly thermoelastic solids. Differential inequality techniques are employed to obtain exponential decay estimates. The results are illustrated by several examples. The results are relevant to Saint-Venant principles for nonlinear thermoelasticity as well as to theorems of Phragmen-Lindelof type.

Spatial Decay Estimates for a Class of Second-Order Quasilinear Elliptic Partial Differential Equations Arising in Anisotropic Nonlinear Elasticity

The spatial decay behavior of solutions of second-order quasilinear elliptic partial differential equations, in divergence form, defined on a two-dimensional semi-infinite strip, is investigated. Such equations arise in the theory of anti-plane shear deformations for anisotropic nonlinearly elastic solids and also in anisotropic nonlinear steady-state heat conduction. Differential inequality techniques are employed to obtain exponential decay estimates. The results are illustrated by several examples, one of which is the (isotropic) minimal surface equation. The results are relevant to Saint-Venant principles for nonlinear elasticity as well as to theorems of Phragmen-Lindelof type.

Solvability of boundary-value problems for quasilinear elliptic and parabolic equations in unbounded domains in classes of functions growing at infinity

For divergent elliptic equations with the natural energetic spaceWpm(Ω),m≥1,p>2, we prove that the Dirichlet problem is solvable in a broad class of domains with noncompact boundaries if the growth of the right-hand side of the equation is determined by the corresponding theorem of Phragmen-Lindelof type. For the corresponding parabolic equation, we prove that the Cauchy problem is solvable for the limiting growth of the initial function % MathType!MTEF!2!1!+- $$u_0 (x) \in L_{2.loc} (R^n ): \int\limits_{|x|< \tau } {u_0^2 dx \leqslant c\tau ^{n + 2mp/(p - 2)} \forall \tau< \infty } $$

Some theorems of Phragmen-Lindelof type for nonlinear partial differential equations

The present paper studies second order partial differential equations in two independent variables of the form Div(?1|u,1|n-1u,1, ?2|u,2|n-1u,2) = 0. We obtain decay estimates for the solutions in a semi-infinite strip. The results may be seen as theorems of Phragmen-Lindelof type. The method is strongly based on the ideas of Horgan and Payne [5], [6], [8].

On the asymptotic behavior of solutions of linear second-order boundary-value problems on a semi-infinite strip

This paper treats the asymptotic behavior of solutions of a linear secondorder elliptic partial differential equation defined on a two-dimensional semiinfinite strip. The equation has divergence form and variable coefficients. Such equations arise in the theory of steady-state heat conduction for inhomogeneous anisotropic materials, as well as in the theory of anti-plane shear deformations for a linearized inhomogeneous anisotropic elastic solid. Solutions of such equations that vanish on the long sides of the strip are shown to satisfy a theorem of Phragmen-Lindelof type, providing estimates for the rate of growth or decay which are optimal for the case of constant coefficients. The results are illustrated by several examples. The estimates obtained in this paper can be used to assess the influence of inhomogeneity and anisotropy on the decay of end effects arising in connection with Saint-Venant's principle.

A Phragmén-Lindelöf theorem

Let Ω be an unbounded and connected domain in E n . Consider on Ω×(0,∞) the parabolic equation u t -div A(x, t, u, ⊇u)=B(x, t, u, ⊇u). Under proper conditions a theorem of Phragmen-Lindelof type is proved for generalized solutions of the equation

GENERALIZATION OF A THEOREM OF MEN'SHOV ON MONOGENIC FUNCTIONS

It is shown that in Men'shov's theorem on the holomorphicity of continuous functions monogenic at each point of a domain with respect to two intervals intersecting at this point the condition of continuity of may be replaced by the condition of summability of for all positive . As a collateral result a theorem of Phragmen-Lindelof type is proved in which a certain summability condition is imposed in place of a condition on the growth of the function. Bibliography: 17 titles.

A variant of the three-ball theorem for the solution of elliptic equations and a related theorem of Phragmen-Lindelof type

A variant of the three-ball theorem for the solution of elliptic equations and a related theorem of Phragmen-Lindelof type

ON AN APPROACH TO THE STUDY OF QUALITATIVE PROPERTIES OF SOLUTIONS OF NONDIVERGENCE ELLIPTIC EQUATIONS OF SECOND ORDER

The author studies some important interior and boundary properties of solutions of linear and quasilinear second order elliptic equations, which may in general be degenerate on the boundary of a (bounded or unbounded) domain. An a priori estimate of the Holder norm of the solution is obtained, and new theorems are proved on regularity of a boundary point with respect to the Dirichlet problem. The Harnack inequality and Phragmen-Lindelof type theorems are also proved.Bibliography: 37 titles.

ON THE ASYMPTOTIC PROPERTIES OF SUBSOLUTIONS OF QUASILINEAR EQUATIONS OF ELLIPTIC TYPE AND MAPPINGS WITH BOUNDED DISTORTION

A theorem of Phragmen-Lindelof type is established for subsolutions of second order quasilinear equations of elliptic type, given in divergence form. Asymptotic properties of entire subsolutions are studied and the results are applied to mappings with bounded distortion. Bibliography: 15 titles.

Teoremi di tipo Liouville e di tipo Phragmen-Lindelöf per le soluzioni di certe equazioni quasiellittiche

Some Liouville type theorems and some Phragmen-Lindelof type theorems for certain non-homogeneous semielliptic boundary value problems are given.

Zuin Phragmen-Lindelofsches Piinzip bei quasiholomorphen und pseudoanalytischen Funktionenf

Quasiholomorphic or quasiregular functions w = u+iv are the solutions of a Beltrami equation Wz = v(z)Wz with v(z) ≤K<1. The relation i(l + v)/(l-v) = G/F defines generating functions Fand G in the sense of L. Berss and the functions W = Fu+Gv are called pseudoanalytic; they are the solutions of the elliptic system Wz = AW+BW with suitable A and B. Sometimes pseudoanalytic functions are useful in proving theorems for quasiholomorphic functions. Here some theorems of Phragmen-Lindelof type are proved. If W{z) is defined in the right half-plane H and |W(z)| is bounded by 1 on the imaginary axis, we have the classical results on the existence of the limits of (log M(r))r, m(r)r, and (log |W(reiγ)r cos γ where M(r) is the supremum on Z= r, m(r) is the mean value of log+ W(z) over = r in H. But if (log M(r))tends to zero we have not the classical result that ∣W(z) is bounded in H. We get only a bound for |W(z)|which depends on the coefficients A and B (or v) and which may increase as rλ for each λ in [0,1). So the Phragmen-Lindelof theorem looses its "gap"charactor.