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.     

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 .     

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.     

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.     

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.     

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.     

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.     

Qualitative behavior of dissipative wave equations on bounded domains

The qualitative behavior of solutions of the mixed problem u_tt = u-a(x)u_t in IR x , u=0 on IR x , is studied in the case when a>0 and IRⁿ is bounded. Roughly speaking, if aa_min>0, then solutions decay at least as fast as exp t( –1/2a_min), with the possible exception of a finite dimensional set of smooth solutions whose existence is associated with a phenomenon of overdamping. If a_max is sufficiently small, depending on , then no overdamping occurs.Partially supported by NSF grant NSF GP 34260.This work was partially supported by the National Science Foundation under Grant No. GP 34260     

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.     

Equivalent Conditions for Ω-Inverse Limit Stability

Recently we obtained sufficient conditions for an endomorphism to be -inverse limit stable. That is, if an endomorphism f satisfies weak Axiom A and the no-cycles condition, then f is -inverse limit stable. In this paper we give alternative conditions for -inverse limit stability. The following are equivalent: (a) f satisfies weak Axiom A and the no-cycles condition; (b) the chain recurrent set is prehyperbolic; and (c) the closure of the set of -limit points of f, L ⁺(f), is prehyperbolic with no cycles.     

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.     

Asymptotic behavior of solutions of nonlinear elliptic equations

We study and obtain formulas for the asymptotic behavior as ¦x¦ of C ² solutions of the semilinear equation u=f(x, u), x (*) where is the complement of some ball in ⁿ and f is continuous and nonlinear in u. If, for large x, f is nearly radially symmetric in x, we give conditions under which each positive solution of (*) is asymptotic, as ¦x¦, to some radially symmetric function. Our results can also be useful when f is only bounded above or below by a function which is radially symmetric in x or when the solution oscillates in sign. Examples when f has power-like growth or exponential growth in the variables x and u usefully illustrate our results.     

Attractor for a Degenerate Nonlinear Diffusion Problem with Nonlinear Boundary Condition

In this paper we prove the existence of a compact attractor in L () for a degenerate nonlinear diffusion problem with nonlinear flux on the boundary. In order to formulate the equation as a dynamical system, some existence and uniqueness results for weak solutions are proved.     

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.     

Extended Korn's inequalities and the associated best possible constants

We are concerned with the coerciveness of the strain energy E(u) (in linear elasticity) associated with a displacement vector u on the Sobolev space H¹ () or its subspaces, a domain in ⁿ representing an isotropic elastic body—certain specific cases are called Korn's inequalities. Sufficient (and necessary) conditions on the Lamé moduli for E(·) to be coercive (or uniformly positive) on such spaces are given, and the associated best possible constants are obtained for some cases.     

On the compactness of quasi-conforming element spaces and the convergence of quasi-conforming element method

In this paper, the compactness of quasi-conforming element spaces and the—convergence of quasi-conforming element method are discussed. The well-known Rellich compactness theorem is generalized to the sequences of quasi-conforming element spaces with certain properties, and the generalized Poincare inequality. The generalized Friedrichs inequality and the generalized inequality of Poincare-Friedrichs are proved true for them. The error estimates are also given. It is shown that the quasi-conforming element method is convergent if the quasi-conforming element spaces have the approximability and the strong continuity, and satisfy the rank condition of element and pass the test IPT. As practical examples, 6-parameter, 9-paramenter, 12-paramenter, 15-parameter, 18-parameter and 21-paramenter quasi-conforming elements are shown to be convergent, and their L²²()-errors are O(h), O(h), O(h ² ), O(h ² ), O(h ), and O(h ⁴ ) respectively.     

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<+.     

Shell vibrations at high-frequency

The free vibration is called high-frequency when the frequency parameter is limited by the inequalities >max{R ₂ ^–2 (s)} and O(h ⁰). In this case there is only one boundary layer type of solution in the neighbourhood of any edge which is not sufficient to satisfy the two non-tangential boundary conditions to be dropped by the membrane equations at the edge, and is called non-complete.An asymptotic approach is presented in this paper, by means of which we find that there are two types of principal modes to be operative over the whole range of the shell surface, when the shell vibrates axisymmetrically at high frequency. One of the principal modes is a membrane type (¦u¦¦w¦, and the index of variation is zero) and the other is a quasi-transverse one with quick variation (¦u¦¦w¦, and the index of variation is equal to 1/2). Correspondingly, the set of frequency parameters can also be divided into two subsets, one of which corresponds to the membrane modes as their eigenvectors, while the other subset corresponds to the quasi-transverse modes with quick variation as their eigenvectors.