Uniqueness and the maximum principle for quasilinear elliptic equations with quadratic growth conditions

In this paper, we show that the maximum principle holds for quasilinear elliptic equations with quadratic growth under general structure conditions.Two typical particular cases of our results are the following. On one hand, we prove that the equation (1) {ie77-01} where {ie77-02} and {ie77-03} satisfies the maximum principle for solutions in H ¹()L(), i.e., that two solutions u ₁, u ₂H¹() L() of (1) such that u ₁u₂ on , satisfy u ₁u₂ in . This implies in particular the uniqueness of the solution of (1) in H ₀ ¹ ()L().On the other hand, we prove that the equation (2) {ie77-04} where fH^–1() and g(u)>0, g(0)=0, satisfies the maximum principle for solutions uH¹() such that g(u)¦Du|{²L¹(). Again this implies the uniqueness of the solution of (2) in the class uH ₀ ¹ () with g(u)¦Du|{²L¹().In both cases, the method of proof consists in making a certain change of function u=(v) in equation (1) or (2), and in proving that the transformed equation, which is of the form (3) {ie77-05}satisfies a certain structure condition, which using ((v₁ -v ₂)⁺)ⁿ for some n>0 as a test function, allows us to prove the maximum principle.     

On the Discretization of Distributed-Parameter Systems with Quadratic and Cubic Nonlinearities

Approximate methods for analyzing the vibrations of an Euler--Bernoulli beam resting on a nonlinear elastic foundation are discussed. The cases of primary resonance ( _n ) and subharmonic resonance of order one-half ( 2 _n ), where is the excitation frequency and _n is the natural frequency of the nth mode of the beam, are investigated. Approximate solutions based on discretization via the Galerkin method are contrasted with direct application of the method of multiple scales to the governing partial-differential equation and boundary conditions. The amplitude and phase modulation equations show that single-mode discretization leads to erroneous qualitative as well as quantitative predictions. Regions of softening (hardening) behavior of the system, the spatial dependence of the response drift, and frequency-response curves are numerically evaluated and compared using both approaches.     

Existence theorem for a minimum problem with free discontinuity set

We study the variational problem Where is an open set in ⁿ ,n2gL ^q () L (), 1q<+, O<, <+ andH _n–1 is the (n–1)-dimensional Hausdorff Measure.     

On a counterexample to a conjecture of Saint Venant

The elliptic boundary value problem % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqGHsi% slcqGHuoarcaWG1bGaeyypa0dccaGae8hiaaIaaGymaiab-bcaGiab% -bcaGiab-bcaGiaabMgacaqGUbGaaeiiaiabfM6axjaabYcaaeaaae% aacaWG1bGaeyypa0JaaGimaiab-bcaGiab-bcaGiab-bcaGiaab+ga% caqGUbGaaeiiaiabgkGi2kabfM6axjaabYcaaaaa!4E11!\[\begin{gathered}- \Delta u = 1 {\text{in }}\Omega {\text{,}} \hfill \\\hfill \\u = 0 {\text{on }}\partial \Omega {\text{,}} \hfill \\\end{gathered}\]is considered. The Saint Venant's conjecture for convex plane domains , having symmetry about two orthogonal axes, is that the maximum of |u| occurs only at the points on which are nearest to the origin. G. Sweers constructed one such domain and claimed that either the conjecture fails for or for ={(x, y);u(x, y) >}, which again is convex. We give a totally different proof of this claim. Our proof brings out clearly the reason for the failure of the conjecture and also allows us to construct many more such domains.     

RADAR ECHO DELAY IN THE(Ω,Aμν)-FIELD THEORY

The present work aims to consider the.fourth test of general relativity theory by Shapiro.using radar echo delay in Yu’s(Ω,A_μν)-field theory.     

The free energy of a macroscopic system

The statistical mechanics of classical and quantum mechanical systems interacting with many-body forces are investigated in the canonical and grand canonical ensembles. Under various general conditions on the attractive and repulsive parts of the potential energy and on the shapes of the domains _k confining the system, it is shown that the canonical free energy per particle and the grand canonical pressure have unique limits for infinite systems which are convex monotonie functions of the specific volume and chemical potential respectively, and satisfy the expected thermodynamic relations.For pure pair forces with potential (r) sufficient conditions are: (r)D ₁/r³⁺ as r0, |(r)|D ₂/r³⁺ as r (>0), and (r)-w₀ all r; the domains _k may be constructed from a finite set of bounded domains of arbitrary shape by any sequence of isotropic expansions such that the volume V(_k) approaches infinity with k.The work reported here was done while the author was on leave of absence from The Wheatstone Physics Laboratory, King's College, London W. C. 2, England.     

Laminar-Turbulent Transition in Spherical Couette Flow for Counter-Rotating Boundaries

The results of an experimental investigation of the transition to stochasticity are given for a thick spherical layer = (r ₂ – r ₁ = 1.006( r ₁ and r ₂ are the radii of the inner and outer spheres). The three-dimensional structure and the velocity fluctuation spectra of the supercritical flow regimes are studied over the range of variation of the Reynolds numbers, calculated on the basis of the parameters of the inner and outer spheres, -950 < Re₂ = ₂ r ₂ ² /v < -700 and 250 < Re₂ = ₂ r ₂ ² /v < 450, where ₂ and ₁ are the angular velocities of rotation of the inner and outer spheres, is the kinematic viscosity of the fluid in the layer, and negative values of Re₂ correspond to rotation of the layer boundaries in opposite directions.     

On the cross-effects of coupled macroscopic transport equations in porous media

In this paper, the derivation of macroscopic transport equations for this cases of simultaneous heat and water, chemical and water or electrical and water fluxes in porous media is presented. Based on themicro-macro passage using the method of homogenization of periodic structures, it is shown that the resulting macroscopic equations reveal zero-valued cross-coupling effects for the case of heat and water transport as well as chemical and water transport. In the case of electrical and water transport, a nonsymmetrical coupling was found.Notations b mobility - c concentration of a chemical - D rate of deformation tensor - D molecular diffusion coefficient - D _ij ^eff macroscopic (or effective) diffusion tensor - electric field - E ₀ initial electric field - k _ij molecular tensor - j, j ^*, current densities - K _ij macroscopic permeability tensor - l characteristic length of the ERV or the periodic cell - L characteristic macroscopic length - L _ijkl coupled flows coefficients - n _i unit outward vector normal to - p pressure - q _t ,q _t ⁺ , heat fluxes - q _c ,q _c ⁺ , chemical fluxes - s specific entropy or the entropy density - S entropy per unit volume - t time variable - t _ij local tensor - T absolute temperature - v _i velocity - V ₀ initial electric potential - V electric potential - x macroscopic (or slow) space variable - y microscopic (or fast) space variable - _i local vectorial field - _i local vectorial field - electric charge density on the solid surface - , bulk and shear viscosities of the fluid - _ij local tensor - _ij local tensor - _i local vector - _ij molecular conductivity tensor - _ij ^eff effective conductivity tensor - homogenization parameter - fluid density - ₀ ion-conductivity of fluid - _ij dielectric tensor - _i ¹ , _i ² , _i ³ local vectors - ⁴ local scalar - _S solid volume in the periodic cell - _L volume of pores in the periodic cell - boundary between _S and _L - ^s rate of entropy production per unit volume - total volume of the periodic cell - _l volume of pores in the cell On leave from the Politechnika Gdanska; ul. Majakowskiego 11/12, 80-952, Gdask, Poland.     

Parabolic Equations in Reifenberg Domains

Natural Sobolev-type estimates are proved for weak solutions of inhomogeneous parabolic equations in divergence form in a bounded cylinder _*=×(0,T] which is -Reifenberg flat in the space direction. The principal coefficients of the operator are assumed to be in BMO space with their BMO semi-norms small enough.     

Injectivity and self-contact in nonlinear elasticity

Let be a bounded open connected subset of ³ with a sufficiently smooth boundary. The additional condition det dx vol () is imposed on the admissible deformations : ¯ of a hyperelastic body whose reference configuration is ¯. We show that the associated minimization problem provides a mathematical model for matter to come into frictionless contact with itself but not interpenetrate. We also extend J. Ball's theorems on existence to this case by establishing the existence of a minimizer of the energy in the space W ^1,p (;³), p > 3, that is injective almost everywhere.     

Lower semicontinuity of the attractor for a singularly perturbed hyperbolic equation

For a smooth, bounded domain R, n 3, and a real, positive parameter, we consider the hyperbolic equationu _tt +u _t –u=–f(u) –g in with Dirichlet boundary conditions. Under certain conditions onf, this equation has a global attractorA inH ₀ ¹ () ×L ²(). For=0, the parabolic equation also has a global attractor which can be naturally embedded into a compact setA ₀ inH ₀ ¹ () ×L ²(). If all of the equilibrium points of the parabolic equation are hyperbolic, it is shown that the setsA are lower semicontinuous at=0. Moreover, we give an estimate of the symmetric distance betweenA ₀ andA .     

Fluctuating flow in a rotating fluid

Summary Fluctuating flow of a viscous fluid rotating over a disk whose angular velocity oscillates about a nonzero mean is investigated. Initially the disk and the fluid rotate in the same sense with different angular velocities ₁ and ₂ ( ₂> ₁) and at a particular instant of time, the angular velocity of the disk becomes ₁[1+ sin( )]. The problem is solved as an initial boundary value problem and it is found that for small values of the results of analytical and numerical methods are in excellent agreement. The effect of frequency parameter on surface skin frictions has been analysed for various values of angular velocity ratio s and amplitude parameter .

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Fluktuierende Strömung in einer rotierenden Flüssigkeit

Übersicht Untersucht wird die fluktuierende Strömung einer viskosen Flüssigkeit, die über einer Scheibe, deren Winkelgeschwindigkeit um einen von Null verschiedenen Mittelwert schwankt, rotiert. Anfangs drehen sich die Scheibe und die Flüssigkeit gleichsinnig, aber mit verschiedenen Winkelgeschwindigkeiten ₁ und ₂ ( ₂> ₁). Zu einem Anfangszeitpunkt geht die Winkelgeschwindigkeit der Scheibe über in ₁[1+ sin ( )]. Die Aufgabe wird als Anfangs-/Randwertproblem gelöst. Für kleine Werte stimmen die analytischen und numerischen Ergebnisse hervorragend überein. Für verschiedene Werte des Winkelgeschwindigkeitsverhältnisses und des Amplitudenparameters wurde der Einfluß des Frequenzparameters auf die Reibspannungen an der Scheibe untersucht.

     

A priori Estimates and Existence for Elliptic Systems via Bootstrap in Weighted Lebesgue Spaces

We present a new general method to obtain regularity and a priori estimates for solutions of semilinear elliptic systems in bounded domains. This method is based on a bootstrap procedure, used alternatively on each component, in the scale of weighted Lebesgue spaces L^p()=L^p((x)dx), where (x) is the distance to the boundary. Using this method, we significantly improve the known existence results for various classes of elliptic systems.     

Instability of stationary solutions for equations of curvature-driven motion of curves

A study is made for equations of evolving curves on a two-dimensional square domain. It is assumed that a curve moves depending on its curvature, normal vector, and position and is orthogonal to at its end points. Under some conditions, instability of stationary solutions is proved through an eigenvalue analysis.     

The chaotic response of a fluttering panel: The influence of maneuvering

The influence of maneuvering on the chaotic response of a fluttering buckled plate on an aircraft has been studied. The governing equations, derived using Lagrangian mechanics, include geometric non-linearities associated with the occurrence of tensile stresses, as well as coupling between the angular velocity of the maneuver and the elastic degrees of freedom. Numerical simulation for periodic and chaotic responses are conducted in order to analyze the influence of the pull-up maneuver on the dynamic behavior of the panel. Long-time histories phase-plane plots, and power spectra of the responses are presented. As the maneuver (load factor) increases, the system exhibits complicated dynamic behavior including a direct and inverse cascade of subharmonic bifurcations, intermittency, and chaos. Beside these classical routes of transition from a periodic state to chaos, our calculations suggest amplitude modulation as a possible new mode of transition to chaos. Consequently this research contributes to the understanding of the mechanisms through which the transition between periodic and strange attractors occurs in, dissipative mechanical systems. In the case of a prescribed time dependent maneuver, a remarkable transition between the different types of limit cycles is presented.Nomenclature a plate length - a _r u _r /h - D plate bending stiffness - E modulus of elasticity - g acceleration due to gravity - h plate thickness - j₁,j₂,j₃ base vectors of the body frame of reference - K spring constant - M Mach number - n 1 + ₀/g - N ₁ applied in-plane force - p–p aerodynamic pressure - P pa ⁴/Dh - q ₀/2 - Q _r generalized Lagrangian forces - R rotation matrix - R ₄ N, a ²/D - t time - kinetic energy - u plate deflection - u displacement of the structure - u _r modal amplitude - v₀ velocity - x coordinates in the inertial frame of reference - z coordinates in the body frame of reference - Ka/(Ka+Eh) - - elastic energy - 2qa ³/D - a/_mh - Poisson's ratio - material coordinates - air density - _m plate density - - _r prescribed functions - _r sin(r z/a) - angular velocity - a/v₀ - skew-symmetric matrix form of the angular velocity     

Effect of latitudinal temperature gradient on spherical Couette flow

The motion of a viscous thermally stratified liquid in a spherical layer due to rotation of the boundaries of the layer at different angular velocities with nonuniform (latitudinal) heating of the outer boundary is investigated. The investigated problem can be regarded as a very simple qualitative model of some astrophysical and geophysical effects, e.g., flows in planetary atmospheres. The characteristic parameters of the problem are the Reynolds number Re=r_i ²_i/, the ratios of the angular velocities of rotation =_e/_i and of the radiia=r_e/r_i of the outer and inner spheres, and the heating nonuniformity parameter . The Boussinesq approximation is adopted in the investigation: It is assumed that the Reynolds numbers Re are fairly small and the solution is sought in the form of a series of whole positive powers of Re. The first two terms of the series are found in analytical form for arbitrary values of the other characteristic parameters. Possible types of meridional flows are established and the regions of the parameters in which a particular type of circulation takes place are determined. It is shown that a latitudinal temperature gradient on the outer boundary leads to the appearance of new (in comparison with the isothermal case) types of meridional flows. The asymptotic form of the stream function of meridional flow in very thin and very thick layers is obtained.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 15–23, March–April, 1975.     

Variants of the Stokes Problem: the Case of Anisotropic Potentials

We investigate the smoothness properties of local solutions of the nonlinear Stokes problem$\begin{eqnarray*}-\diverg \{T(\eps(v))\} + \nabla \pi &=& g \msp \mbox{on $\Omega$,}\\\diverg v&\equiv & 0 \msp \mbox{on $\Omega$,}\end{eqnarray*}$where v: ⁿ is the velocity field, $\pi$: $ denotes the pressure function, and g: ⁿ represents a system of volume forces, denoting an open subset of ⁿ . The tensor T is assumed to be the gradient of some potential f acting on symmetric matrices. Our main hypothesis imposed on f is the existence of exponents 1 < p q < \infty such that\lambda (1+|\eps|^{2})^{\frac{p-2}{2}} |\sigma|^{2} \leq D^{2}f(\eps)(\sigma ,\sigma) \leq \Lambda (1+|\eps|^{2})^{\frac{q-2}{2}} |\sigma|^{2}holds with suitable constants , > 0, i.e. the potential f is of anisotropic power growth. Under natural assumptions on p and q we prove that velocity fields from the space W ¹ _{p, loc} (; ⁿ ) are of class C ^1, on an open subset of with full measure. If n = 2, then the set of interior singularities is empty.Dedicated to O. A. Ladyzhenskaya on the occasion of her 80th birthday     

Bestimmung der Schubspannungsfunktion des Blutes mit dem Couette-Rheometer unter Berücksichtigung des Wandverhaltens

Zusammenfassung Bei Blutergeben sich aus Messungen an verschiedenen Couette-Systemen verschiedene Verläufe für die Schubspannungsfunktion, wenn man Wandhaften annimmt. Es wird daher ein Wandeffekt angenommen, bei dem die unentmischte Blutsuspension unmittelbar auf der Wand gleitet. Die Wandgleitgeschwindigkeit wird als Funktion der Wandschubspannung angesetzt und aus Messungen an drei verschiedenen Couette-Systemen bestimmt.Aus der so ermittelten Wandgleitgeschwindigkeit kann die Dicke eines Blutplasmafilmes an der Wand abgeschätzt werden. Sie ergibt sich zu einigen µm. Dadurch wird die angenommene Modellvorstellung für den Wandeffekt bestätigt.Bei Berücksichtigung der ermittelten Wandgleitgeschwindigkeit ergibt sich für die Schubspannungsfunktion aus Messungen an verschiedenen Couette-Systemen derselbe Verlauf. Bei Annahme von Wandhaften ergeben sich dagegen deutlich zu hohe Werte für die Schubspannungsfunktion.

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Summary Measurements with blood in different Couette-Systems are resulting in different shear functionsf() if no wall-slip effect is assumed.Therefore we use the model that the homogeneous blood suspension is sliding directly on the wall. The wall-slip velocity is introduced as a function of the wall shear stress. This wall-slip function can be determined from measurements with three different Couette-Systems.After the wall-slip function is determined the thickness of a plasma film on the wall can be estimated. One gets a thickness of a few µms. Thus the assumed model for the wall effect is confirmed. Measurements with different Couette-Systems evaluated according to the wall-slip model, are leading to the same shear functionf().

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D Rohrdurchmesser - f() Schubspannungsfunktion - f() mittlerer Fehler vonf() - h Länge des Couette-Spaltes - M in der Couette-Strömung übertragenes Drehmoment - M _ij Meßwerte fürM - Mittelwert derM _ij - mittlerer Fehler von - r _a ,r _i äußerer bzw. innerer Begrenzungsradius eines Couette-Systems - r ₁,r ₂,r ₃ Begrenzungsradien der Couette-Systeme I, II, III - s Spaltweite im Couette-System - v _w Wandgleitgeschwindigkeit - Schergeschwindigkeit - Dicke des Wandfilms - ( _w) Funktion des Wandgleitens - Viskosität - Schubspannungsfeld im Couette-System - ₁, ₂, ₃ Schubspannungen an den Stellenr ₁,r ₂,r ₃ in den Couette-Systemen I, II, III bei gleichem Moment - _w Wandschubspannung - Winkelgeschwindigkeitsdifferenz zwischen äußerem und innerem Zylinder im Couette-System - _i vorgegebene Werte für - mittlerer Fehler von - _I, _II, _III für die Couette-Systeme I, II, III - _{I, II, III} mittlere Fehler von _I, _II, _III Vortrag, gehalten auf der Jahrestagung der Deutschen Rheologischen Gesellschaft in Aachen vom 5.–7. März 1979.Mit 5 Abbildungen     

How Violent are Fast Controls for Schrödinger and Plate Vibrations?

Given a time T>0 and a region on a compact Riemannian manifold M, we consider the best constant, denoted C_T,, in the observation inequality for the Schrödinger evolution group of the Laplacian with Dirichlet boundary condition: We investigate the influence of the geometry of on the growth of C_T, as T tends to 0.By duality, C_T, is also the controllability cost of the free Schrödinger equation on M with Dirichlet boundary condition in time T by interior controls on . It relates to hinged vibrating plates as well. We analyze separately the effects of wavelengths which are greater and lower than the order of the control time T. We emphasize a tool of wider scope: the control transmutation method.We prove that C_T, grows at least like exp(d²/4T), where d is the largest distance of a point in M from , and at most like exp(_*L²/T), where L is the length of the longest generalized geodesic in M which does not intersect , and _* ]0,4[ is the best constant in the following inequality for the Schrödinger equation on the segment [0,L] observed from the left end: where A is the operator _x² with domain D(A)={fH²(0,L),|,Bf(0)=0=f(L)} and the inequality holds with B=1 and with B=_x. We also deduce such upper bounds on product manifolds for some control regions which are not intersected by all geodesics.     

Stability of MHD Couette flow in a narrow gap annulus

A numerical solution to the MHD stability problem for dissipative Couette flow in a narrow gap is presented under following conditions: (i) the inner cylinder rotating with the outer one stationary, (ii) co-rotating cylinders, (iii) counter-rotating cylinders, (iv) an axially applied magnetic field, and (v) conducting and non-conducting walls.Results for the critical wave number and the critical Taylor number are presented and are compared with those of Chandrasekhar (1953). The agreement is very good. The amplitude of the radial velocity and the cell-pattern are shown on graphs for both the conducting and non-conducting walls and for different values of ± (=₂/₁, ₂-the angular velocity of the outer cylinder, ₁-the angular velocity of the inner cylinder) and Q the magnetic field parameter which is the square of the Hartman number. The effects of ± and Q on the stability of the flow are discussed. It is seen that the effect of the magnetic field is to inhibit the onset of instability, it being more so in the presence of conducting walls than in the presence of non-conducting walls.