Mathematische Annalen最新文献

The eventual concavity properties are useful to characterize geometric properties of the final state of solutions to parabolic equations. In this paper we give characterizations of the eventual concavity properties of the heat flow for nonnegative, bounded measurable initial functions with compact support.

In this paper we study existence and nonexistence of positive radial solutions of a Dirichlet problem for the prescribed mean curvature operator with weights in a ball with a suitable radius. Because of the presence of different weights, possibly singular or degenerate, the problem under consideration appears rather delicate, it requires an accurate qualitative analysis of the solutions, as well as the use of Liouville type results based on an appropriate Pohozaev type identity. In addition, sufficient conditions for global solutions to be oscillatory are given.

Let ((h_I)) denote the standard Haar system on [0, 1], indexed by (Iin mathcal {D}), the set of dyadic intervals and (h_Iotimes h_J) denote the tensor product ((s,t)mapsto h_I(s) h_J(t)), (I,Jin mathcal {D}). We consider a class of two-parameter function spaces which are completions of the linear span (mathcal {V}(delta ^2)) of (h_Iotimes h_J), (I,Jin mathcal {D}). This class contains all the spaces of the form X(Y), where X and Y are either the Lebesgue spaces (L^p[0,1]) or the Hardy spaces (H^p[0,1]), (1le p < infty ). We say that (D:X(Y)rightarrow X(Y)) is a Haar multiplier if (D(h_Iotimes h_J) = d_{I,J} h_Iotimes h_J), where (d_{I,J}in mathbb {R}), and ask which more elementary operators factor through D. A decisive role is played by the Capon projection (mathcal {C}:mathcal {V}(delta ^2)rightarrow mathcal {V}(delta ^2)) given by (mathcal {C} h_Iotimes h_J = h_Iotimes h_J) if (|I|le |J|), and (mathcal {C} h_Iotimes h_J = 0) if (|I| > |J|), as our main result highlights: Given any bounded Haar multiplier (D:X(Y)rightarrow X(Y)), there exist (lambda ,mu in mathbb {R}) such that

i.e., for all (eta > 0), there exist bounded operators A, B so that AB is the identity operator ({{,textrm{Id},}}), (Vert AVert cdot Vert BVert = 1) and (Vert lambda mathcal {C} + mu ({{,textrm{Id},}}-mathcal {C}) - ADBVert < eta ). Additionally, if (mathcal {C}) is unbounded on X(Y), then (lambda = mu ) and then ({{,textrm{Id},}}) either factors through D or ({{,textrm{Id},}}-D).

We study the fractional p-Laplace equation

for (0<s<1) and in the subquadratic case (1<p<2). We provide Hölder estimates with an explicit Hölder exponent. The inhomogeneous equation is also treated and there the exponent obtained is almost sharp for a certain range of parameters. Our results complement the previous results for the superquadratic case when (pge 2). The arguments are based on a careful Moser-type iteration and a perturbation argument.

We extend anti-classification results in ergodic theory to the collection of weakly mixing systems by proving that the isomorphism relation as well as the Kakutani equivalence relation of weakly mixing invertible measure-preserving transformations are not Borel sets. This shows in a precise way that classification of weakly mixing systems up to isomorphism or Kakutani equivalence is impossible in terms of computable invariants, even with a very inclusive understanding of “computability”. We even obtain these anti-classification results for weakly mixing area-preserving smooth diffeomorphisms on compact surfaces admitting a non-trivial circle action as well as real-analytic diffeomorphisms on the 2-torus.

We investigate the behaviour of radial solutions to the Lin–Ni–Takagi problem in the ball (B_R subset mathbb {R}^N) for (N ge 3):

when p is close to the first critical Sobolev exponent (2^* = frac{2N}{N-2}). We obtain a complete classification of finite energy radial smooth blowing up solutions to this problem. We describe the conditions preventing blow-up as (p rightarrow 2^*), we give the necessary conditions in order for blow-up to occur and we establish their sharpness by constructing examples of blowing up sequences. Our approach allows for asymptotically supercritical values of p. We show in particular that, if (p ge 2^*), finite-energy radial solutions are precompact in (C^2(overline{B_R})) provided that (Nge 7). Sufficient conditions are also given in smaller dimensions if (p=2^*). Finally we compare and interpret our results in light of the bifurcation analysis of Bonheure, Grumiau and Troestler in (Nonlinear Anal 147:236–273, 2016).

We consider the physical setup of a three-dimensional fluid–structure interaction problem. A viscous compressible gas or liquid interacts with a nonlinear, visco-elastic, three-dimensional bulk solid. The latter is described by an evolution with inertia, a non-linear dissipation term and a term that relates to a non-convex elastic energy functional. The fluid is modelled by the compressible Navier–Stokes equations with a barotropic pressure law. Due to the motion of the solid, the fluid domain is time-changing. Our main result is the long-time existence of a weak solution to the coupled system until the time of a collision. The nonlinear coupling between the motions of the two different matters is established via the method of minimising movements. The motion of both the solid and the fluid is chosen via an incrimental minimization with respect to dissipative and static potentials. These variational choices together with a careful construction of an underlying flow map for our approximation then directly result in the pressure gradient and the material time derivatives.

Let Y be a prequantization bundle over a closed spherically monotone symplectic manifold (Sigma ). Adapting an idea due to Diogo and Lisi, we study a split version of Rabinowitz Floer homology for Y in the following two settings. First, (Sigma ) is a symplectic hyperplane section of a closed symplectic manifold X satisfying a certain monotonicity condition; in this case, (X {{setminus }} Sigma ) is a Liouville filling of Y. Second, the minimal Chern number of (Sigma ) is greater than one, which is the case where the Rabinowitz Floer homology of the symplectization (mathbb {R} times Y) is defined. In both cases, we construct a Gysin-type exact sequence connecting the Rabinowitz Floer homology of (X{setminus }Sigma ) or (mathbb {R} times Y) and the quantum homology of (Sigma ). As applications, we discuss the invertibility of a symplectic hyperplane section class in quantum homology, the isotopy problem for fibered Dehn twists, the orderability problem for prequantization bundles, and the existence of translated points. We also provide computational results based on the exact sequence that we construct.

The aim of this paper is to study singular solutions for a one-dimensional nonlinear diffusion equation. Due to slow diffusion near singular points, there exists a solution with a singularity at a prescribed position depending on time. To study properties of such singular solutions, we define a minimal singular solution as a limit of a sequence of approximate solutions with large Dirichlet data. Applying the comparison principle and the intersection number argument, we discuss the existence and uniqueness of a singular solution for an initial-value problem, the profile near singular points and large-time behavior of solutions. We also give some results concerning the appearance of a burning core, convergence to traveling waves and the existence of an entire solution.