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 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.     

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     

An experimental study on vortex breakdown in a differentially-rotating cylindrical container

The vortex breakdown phenomenon in a closed cylindrical container with a rotating endwall disk was reproduced. Visualizations were performed to capture the prominent flow characteristics. The locations of the stagnation points of breakdown bubbles and the attendant global flow features were in excellent agreement with the preceding observations. Experiments were also carried out in a differentially-rotating cylindrical container in which the top endwall rotates at a relatively high angular velocity _t, and the bottom endwall and the sidewall rotate at a low angular velocity _sb. For a fixed cylinder aspect ratio, and for a given relative rotational Reynolds number based on the angular velocity difference _t–_sb, the flow behavior is examined as |_sb/_t| increases. For a co-rotation (_sb/_t>0), the breakdown bubble is located closer to the bottom endwall disk. However, for a counter-rotation (_sb/_t<0), the bubble is seen closer to the top endwall disk. For sufficiently large values of _sb, the bubble ceases to exist for both cases.     

Uniqueness theorems for displacement fields with locally finite energy in linear elastostatics

Uniquencess theorems are proved for the fundamental boundary value problems of linear elastostatics in bodies of arbitrary shape. The displacement fields are required to have finite strain energy in bounded portions of the bodies and satisfy the principle of virtual work. For bounded bodies, the total strain energy is finite and uniquencess is proved without additional hypotheses. In particular, no restrictions other than the energy condition are placed on the field singularities that may occur at sharp edges and corners. For unbounded bodies, uniqueness can be proved as in the bounded case if the total strain energy is finite. Sufficient conditions for this are shown to be the finiteness of the strain energy in bounded portions of the body together with the growth restriction % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqef0uAJj3BZ9Mz0bYu% H52CGmvzYLMzaerbd9wDYLwzYbItLDharqqr1ngBPrgifHhDYfgasa% acOqpw0xe9v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8Wq% Ffea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qqQ8frFve9Fve9Ff0dme% GabaqaaiGacaGaamqadaabaeaafiaakeaadaWdraqaaiaabwhadaWg% aaWcbaGaaeyAaaqabaGccaGGOaGaaeiEaiaacMcacaqG1bWaaSbaaS% qaaiaabMgaaeqaaOGaaiikaiaabIhacaGGPaGaaeizaiaabIhacaqG% 9aGaaGimaiaacIcacaqGYbGaaiykaiaacYcacaqGYbGaeyOKH4Qaey% OhIukaleaacqGHPoWvdaWgaaadbaGaaeOCaiaacYcacqaH0oazaeqa% aaWcbeqdcqGHRiI8aaaa!5E73!\[\int_{\Omega _{{\text{r}},\delta } } {{\text{u}}_{\text{i}} ({\text{x}}){\text{u}}_{\text{i}} ({\text{x}}){\text{dx = }}0({\text{r}}),{\text{r}} \to \infty } \] on the displacement fieldu _i , where _r, is the portion of the body that lies between concentric spheres with radiir andr+ and >0.This research was supported by the Air Force Office of Scientific Research. Reproduction in whole or part is permitted for any purpose of the United States Government.Prepared under Contract No. F 49620-77-C-0053 for Air Force Office of Scientific Research.     

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.

     

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 .     

Resonant generation of a solitary wave in a thermocline

The resonant generation of a second-mode internal solitary wave, resulting from a ship internal waves system damping in a thermocline, is studied experimentally. The source of the stationary internal waves was provided by an oblong ellipsoid of revolution towed horizontally and uniformly at the depth of the thermocline center. The ranges of the Reynolds and Froude numbers were 500Re=Ul/v 15000 and 0.3Fi=U/N _max D1.0, respectively. When the body's speed and the linear long-wave second-mode phase speed were equal, an internal solitary wave of the bulge type was observed. The shape of the wave satisfied the Korteweg-de Vries equation. The Urcell parameter was equal to 10.2.List of Symbols L, B, H towing tank length, breadth and height respectively - z vertical coordinate - D characteristic vertical dimension of the body - a minor semiaxis of an ellipsoid - b major semiaxis of an ellipsoid (maximum ellipsoid diameter D=2a) - l length of the body ( =2b) - U velocity of the body - t temperature - g acceleration due to gravity - i fresh water density at ith level - _4° fresh water density for temperature t=4°C - o water density at the center of the thermocline - _i density variation due to the temperature variation at the ith horizon - N Brunt-Väisälä frequency - N _max maximum value of Brunt-Väisälä frequency - Re Reynolds number - Fi internal Froude number - f _n eigenfunction of the boundary-value problem for the nth mode - _n nth mode frequency - k _{n n}th mode horizontal wavenumber - C _n limiting phase speed of a linear nth mode interval wave (= _n/k_n;k_n 0) - Ur Urcell parameter - v fresh water kinematic viscosity - conventional density - half-length of a solitary wave - ₀ solitary wave height - time This work was partially supported by the INTAS (grant no. 94-4057) and by the Russian Foundation of Basic Research under grant no. 94-05-17004-a.A version of this paper was presented at the Second International Conference on Experimental Fluid Mechanics, Torino, Italy, 4–8 July, 1994.     

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.     

On motions which preserve ellipsoidal holes

Let be an ellipsoid in ³ contained in a region . Suppose one body occupies the region – in a certain stress-free reference configuration while a second body, the inclusion, occupies the region in a stress-free reference configuration. Assume the inclusion is free to slip at . Now suppose that by changing some variable such as the temperature, pressure, humidity, etc., we cause the trivial deformation y(x)=x of the inclusion to become unstable relative to some other deformation. For example, the inclusion may be made out of such a material that if it were removed from the body, it would suddenly change shape to another stress-free configuration specified by a deformation y=Fx, F F=C, C being a fixed tensor characteristic of the material, at a certain temperature. However, with an appropriate material model for the surrounding body, we expect it will resist the transformation, and both body and inclusion will end up stressed.In a recent paper, Mura and Furuhashi [1] find the following unexpected result within the context of infinitesimal deformations: certain homogeneous deformations of the ellipsoid which take it to a stress-free configuration also leave the surrounding body stress-free. These are essentially homogeneous, infinitesimal deformations which preserve ellipsoidal holes. In this paper, we find all finite homogeneous deformations and motions which preserve ellipsoidal holes.     

Influence of rotation of the body and exterior vorticity on the heat transfer near the stagnation point of a blunt body in a supersonic stream

A study is made of the influence of longitudinal vorticity on the hypersonic viscous shock layer near an axisymmetric cooled surface that rotates about the longitudinal axis with angular velocity ₁ [1, 2]. The equations of the viscous shock layer for the neighborhood of the stagnation point are simplified on the basis of the theory of a thin three-dimensional shock layer [1, 2]. The results are given of some calculations of the influence of the parameters and ₁ on the heat transfer and the structure of the shock layer in the case of steady flow. An iterative numerical method proposed by the author [1] is used, and a modification to accelerate the convergence of the iterations is proposed. It is noted that the parameters and ₁ have characteristic ranges of variation, which depend on the Mach and Reynolds numbers, for which the distributions of the streamlines in the shock layer are qualitatively different.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 179–182, May–June, 1984.I thank V. Ya. Neiland and Yu. D. Shevelev for helpful discussions of this paper.     

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.     

Scalar Minimizers with Fractal Singular Sets

Lack of regularity of local minimizers for convex functionals with non-standard growth conditions is considered. It is shown that for every >0 there exists a function aC() such that the functional admits a local minimizer uW^1,p() whose set of non-Lebesgue points is a closed set with dim()>N–p–, and where 1<p<N<N+<q<+.     

The Extinction Behavior of the Solutions for a Class of Reaction-Diffusion Equations

1　ＩｎｔｒｏｄｕｃｔｉｏｎａｎｄＬｅｍｍａｓＴｈｅｒｅａｒｅｍａｎｙｒｅｓｕｌｔｓａｂｏｕｔｅｘｉｓｔｅｎｃｅ (ｇｌｏｂａｌｏｒｌｏｃａｌ)ａｎｄａｓｙｍｐｔｏｔｉｃｂｅｈａｖｉｏｒｏｆｓｏｌｕｔｉｏｎｓｆｏｒｒｅａｃｔｉｏｎ＿ｄｉｆｆｕｓｉｏｎｅｑｕａｔｉｏｎｓ[1- 9].Ｂｙｔｈｅａｉｄｓｏｆｒｅｓｕｌｔｓ[2 ,3]ｏｆｅｑｕａｔｉｏｎ ｕ/ ｔ=Δｕ-λ｜ｕ｜γ- 1ｕｗｉｔｈｉｎｉｔｉａｌ＿ｂｏｕｎｄａｒｙｖａｌｕｅｓ,ｐａｐｅｒ [4 ]ｓｔｕｄｉｅｄｔｈｅｐｒｏｂｌｅｍｏｆ ｕ/ ｔ=Δｕ-λ｜ｅβｔｕ｜γ- …     

Microstructures with finite surface energy: the two-well problem

We study solutions of the two-well problem, i.e., maps which satisfy uSO(n)ASO(n)B a.c. in ⁿ , where A and B are n×n matrices with positive determinants. This problem arises in the study of microstructure in solid-solid phase transitions. Under the additional hypothesis that the set E where the gradient lies in SO(n) A has finite perimeter, we show that u is locally only a function of one variable and that the boundary of E consists of (subsets of) hyperplanes which extend to and which do not intersect in . This may not be the case if the assumption on E is dropped. We also discuss applications of this result to magnetostrictive materials.     

Waves in a plasma with Hall dispersion

In a magnetohydrodynamic approximation, an investigation is made of the propagation of waves in a plasma, whose characteristic frequency is much less than the collision frequency of the electrons _e ^–1. It is assumed that the magnetic field is sufficiently strong so that the equality _ee1 will be satisfied, where _e is the cyclotron frequency of the rotation of the electrons. With large magnetic Reynolds numbers (R_m1), which are characteristic for many astrophysical problems, this latter condition leads to a need to take account of dispersion effects connected with Hall currents, in the absence of Joule dissipation. The dispersion equation for the propagation of small perturbations is analyzed in the limiting cases of weak dispersion and of a wave propagating along the magnetic field. In the case of weak dispersion, an equation is derived for nonlinear waves. The solutions are found in the form of stationary solitons. The region of such solutions is analyzed. A typical example of a medium with Hall dispersion is an interplanetary plasma, in which the parameter _ee is generally great.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 108–113, May–June, 1974.     

The value of the critical exponent for reaction-diffusion equations in cones

Let D R ^N be a cone with vertex at the origin i.e., D = (0, )x where S ^N–1 and x D if and only if x = (r, ) with r=¦x¦, . We consider the initial boundary value problem: u _t = u+u ^p in D×(0, T), u=0 on Dx(0, T) with u(x, 0)=u ₀(x) 0. Let ₁ denote the smallest Dirichlet eigenvalue for the Laplace-Beltrami operator on and let ₊ denote the positive root of (+N–2) = ₁. Let p ^* = 1 + 2/(N + ₊). If 1 < p < p ^*, no positive global solution exists. If p>p ^*, positive global solutions do exist. Extensions are given to the same problem for u _t=+¦x¦ u ^p .This research was supported in part by the Air Force Office of Scientific Research under Grant # AFOSR 88-0031 and in part by NSF Grant DMS-8 822 788. The United States Government is authorized to reproduce and distribute reprints for governmental purposes not withstanding any copyright notation therein.     

An existence result for a class of shape optimization problems

Given a bounded open subset of R ⁿ, we prove the existence of a minimum point for a functional F defined on the family A() of all quasiopen subsets of , under the assumption that F is decreasing with respect to set inclusion and that F is lower semicontinuous on A() with respect to a suitable topology, related to the resolvents of the Laplace operator with Dirichlet boundary condition. Applications are given to the existence of sets of prescribed volume with minimal k ^th eigenvalue (or with minimal capacity) with respect to a given elliptic operator.     

Flow visualization of compressible vortex structures using density gradient techniques

Mathematical results are derived for the schlieren and shadowgraph contrast variation due to the refraction of light rays passing through two-dimensional compressible vortices with viscous cores. Both standard and small-disturbance solutions are obtained. It is shown that schlieren and shadowgraph produce substantially different contrast profiles. Further, the shadowgraph contrast variation is shown to be very sensitive to the vortex velocity profile and is also dependent on the location of the peak peripheral velocity (viscous core radius). The computed results are compared to actual contrast measurements made for rotor tip vortices using the shadowgraph flow visualization technique. The work helps to clarify the relationships between the observed contrast and the structure of vortical structures in density gradient based flow visualization experiments.Nomenclature a Unobstructed height of schlieren light source in cutoff plane, m - c Blade chord, m - f Focal length of schlieren focusing mirror, m - C _T Rotor thrust coefficient, T/( ² R ⁴) - I Image screen illumination, Lm/m ² - l Distance from vortex to shadowgraph screen, m - n _b Number of blades - p Pressure,N/m ² - p Ambient pressure, N/m ² - r, , z Cylindrical coordinate system - r _c Vortex core radius, m - Non-dimensional radial coordinate, (r/r _c ) - R Rotor radius, m - Tangential velocity, m/s - Specific heat ratio of air - Circulation (strength of vortex), m ²/s - Non-dimensional quantity, ² 8²p r _c ² - Refractive index of fluid medium - ₀ Refractive index of fluid medium at reference conditions - Gladstone-Dale constant, m ³/kg - Density, kg/m ³ - Density at ambient conditions, kg/m ³ - Non-dimensional density, (/ ) - Rotor solidity, (n _b c/ R) - Rotor rotational frequency, rad/s     

Reduced and Generalized Stokes Resolvent Equations in Asymptotically Flat Layers, Part I: Unique Solvability

We study the generalized Stokes equations in asymptotically flat layers, which can be considered as compact perturbations of an infinite (flat) layer Besides standard non-slip boundary conditions, we consider a mixture of slip and non-slip boundary conditions on the upper and lower boundary, respectively. In the first part we prove the unique solvability in L^q-Sobolev spaces, 1 < q < , by extending the known results in the case of an infinite layer ₀ via a perturbation argument to asymptotically flat layers which are sufficiently close to ₀. Combining this result with standard cut-off techniques and the parametrix constructed in the second part, we prove the unique solvability for an arbitrary asymptotically flat layer. Moreover, we show equivalence of unique solvability of the generalized and the reduced Stokes resolvent equations, which is essential for the second part of this contribution.