We show the existence of infinitely many admissible weak solutions for the incompressible porous media equations for all Muskattype initial data with C 3,α -regularity of the interface in the unstable regime and for all non-horizontal data with C 3,α -regularity in the stable regime. Our approach involves constructing admissible subsolutions with piecewise constant densities. This allows us to give a rather short proof where it suffices to calculate the velocity and acceleration at time zero -thus emphasizing the instantaneous nature of non-uniqueness due to discontinuities in the initial data.
We consider a linear elastic plate with stress-free boundary conditions in the limit of vanishing Poisson coefficient. We prove that under a local change of Young's modulus infinitely many eigenvalues arise in the essential spectrum which accumulate at a positive threshold. We give estimates on the accumulation rate and on the asymptotical behaviour of the eigenvalues.
ABSTRACT. We derive Lieb-Thirring inequalities for the Riesz means of eigenvalues of order γ≥3/4 for a fourth order operator in arbitrary dimensions. We also consider some extensions to polyharmonic operators, and to systems of such operators, in dimensions greater than one. For the critical case γ=1−1/(2l) in dimension d=1 with l≥2 we prove the inequality L where V is a real-valued function. For suitable V the negative spectrum of this operator is discrete. The Lieb-Thirring inequalities are estimates on the negative eigenvalues of the form 1which holds for certain γ ≥ 0 with a constant L l,γ,d , depending only on l, d and γ. Here and in the following we use the abbreviationsThis type of inequalities was introduced by Lieb and Thirring in [15]. They proved that (0.1) holds in the case l = 1 for all γ > max(0, ν) with a finite constant L l,γ,d . Their argument can easily be extended to all l ≥ 1. On the other hand it is known that (0.1) fails for γ = 0 if d = 2l and for 0 ≤ γ < ν if d < 2l. In the critical case γ = 0, d > 2l the bound (0.1) exists and is for l = 1 known as the Cwikel-LiebRosenblum inequality, see [4,14,20] and also [3,13]. The existence of L l,γ,d in the remaining critical case d < 2l, γ = ν was verified by Netrusov and Weidl for integer values of l in [21,19]. Hence, the cases of existence for bounds of type (0.1) with γ ≥ 0 are completely settled for integer l, while for non-integer l only the case 2l > d, γ = ν is still open.1991 Mathematics Subject Classification. Primary 35P15; Secondary 47A75, 35J10. 1 Here and below we use the notion 2x− := |x| − x for the negative part of variables, functions, Hermitian matrices or self-adjoint operators.
Abstract. For an infinite linear elastic plate with stress-free boundary, the trapped modes arising around holes in the plate are investigated. These are L 2 -eigenvalues of the elastostatic operator in the punched plate subject to Neumann type stress-free boundary conditions at the surface of the hole. It is proved that the perturbation gives rise to infinitely many eigenvalues embedded into the essential spectrum. The eigenvalues accumulate to a positive threshold. An estimate of the accumulation rate is given. §1. Introduction We consider a homogeneous and isotropic linear elastic medium in the domainThen the quadratic forminduces the elastostatic operator A for materials with zero Poisson ratio; A corresponds to the differential expressionwith stress-free (Neumann-type) boundary conditions. 1 The operator A has purely absolutely continuous spectrum filling the nonnegative half-line. Now, consider a punched plate Ω c = G \ Ω with a hole Ω = Ω 0 × J, where Ω 0 ⊂ R 2 is a bounded Lipschitz domain. Let A Ω c be the elastostatic operator corresponding to the differential expression (1.2) on the outer domain Ω c subject to stress-free (Neumanntype) boundary conditions. This geometric perturbation does not change the location of the essential spectrum, but it gives rise to a somewhat unexpected trapping effect. As the main result of this paper we prove the existence of infinitely many eigenvalues {ν k } k∈N of A Ω c , embedded into the essential spectrum, which accumulate at a certain threshold Λ > 0, and we compute the accumulation rate of these trapped modes. We establish the formula2010 Mathematics Subject Classification. Primary 74B05. Key words and phrases. Elasticity operator, trapped modes. 1 Here we have chosen a suitable set of units such that Young's modulus E fulfills E = 2. Moreover, we point out that the special choice of zero Poisson ratio is essential for the results of this paper.
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