Calculus of Variations and Partial Differential Equations最新文献

We consider the Cauchy problem for the isentropic compressible Euler–Maxwell equations under general pressure laws in a three-dimensional periodic domain. For any smooth initial electron density away from the vacuum and smooth equilibrium-charged ion density, we could construct infinitely many (alpha )-Hölder continuous entropy solutions emanating from the same initial data for (alpha <frac{1}{7}). Especially, the electromagnetic field belongs to the Hölder class (C^{1,alpha }). Furthermore, we provide a continuous entropy solution satisfying the entropy inequality strictly. The proof relies on the convex integration scheme. Due to the constrain of the Maxwell equations, we propose a method of Mikado potential and construct new building blocks.

We prove an analogue statement to an estimate by De Lellis–Müller in (mathbb {R}^3) on the standard Minkowski lightcone. More precisely, we show that under some additional assumptions, any spacelike cross section of the standard lightcone is (W^{2,2})-close to a round surface provided the trace-free part of a scalar second fundamental form A is sufficiently small in (L^2). To determine the correct intrinsically round cross section of reference, we define an associated 4-vector, which transforms equivariantly under Lorentz transformations in the restricted Lorentz group. A key step in the proof consists of a geometric, scaling invariant estimate, and we give two different proofs. One utilizes a recent characterization of singularity models of null mean curvature flow along the standard lightcone by the author, while the other is heavily inspired by an almost-Schur lemma by De Lellis–Topping.

A level orbit of a mechanical Hamiltonian system is a solution of Newton equation that is contained in a level set of the potential energy. In 2003, Mark Levi asked for a characterization of the smooth potential energy functions on the plane with the property that any point on the plane lies on a level orbit; we call such functions Levi potentials. The basic examples are the radial monotone increasing smooth functions. In this paper we show that any Levi potential that is analytic or has totally path-disconnected critical set must be radial. Nevertheless, we show that every compact convex subset of the plane is the critical set of a Levi potential. A crucial observation for these theorems is that, outside the critical set, the family of level sets of a Levi potential forms a solution of the inverse curvature flow.

In this paper, we consider Ricci flows admitting closed and smooth tangent flows in the sense of Bamler (Structure theory of non-collapsed limits of Ricci flows, 2020. arXiv:2009.03243). The tangent flow in question can be either a tangent flow at infinity for an ancient Ricci flow, or a tangent flow at a singular point for a Ricci flow developing a finite-time singularity. Among other things, we prove: (1) that in these cases the tangent flow must be unique, (2) that if a Ricci flow with finite-time singularity has a closed singularity model, then the singularity is of Type I and the singularity model is the tangent flow at the singular point; this answers a question proposed in Chow et al. (The Ricci flow: techniques and applications. Part III. Geometric-analytic aspects. Mathematical surveys and monographs, vol 163. AMS, Providence, 2010), (3) a dichotomy theorem that characterizes ancient Ricci flows admitting a closed and smooth backward sequential limit.

We show that any minimizer of the well-known ACF functional (for the p-Laplacian) constitutes a viscosity solution. This allows us to establish a uniform flatness decay at the two-phase free boundary points to improve the flatness, which boils down to (C^{1,eta }) regularity of the flat part of the free boundary. This result, in turn, is used to prove the Lipschitz regularity of minimizers by a dichotomy argument. It is noteworthy that the analysis of branch points is also included.

In this paper we present the homogenization for nonlinear viscoelastic second-grade non-simple perforated materials at large strain in the quasistatic setting. The reference domain (Omega _{varepsilon }) is periodically perforated and is depending on the scaling parameter (varepsilon ) which describes the ratio between the size of the whole domain and the small periodic perforations. The mechanical energy depends on the gradient and also the second gradient of the deformation, and also respects positivity of the determinant of the deformation gradient. For the viscous stress we assume dynamic frame indifference and it is therefore depending on the rate of the Cauchy-stress tensor. For the derivation of the homogenized model for (varepsilon rightarrow 0) we use the method of two-scale convergence. For this uniform a priori estimates with respect to (varepsilon ) are necessary. The most crucial part is to estimate the rate of the deformation gradient. Due to the time-dependent frame indifference of the viscous term, we only get coercivity with respect to the rate of the Cauchy-stress tensor. To overcome this problem we derive a Korn inequality for non-constant coefficients on the perforated domain. The crucial point is to verify that the constant in this inequality, which is usually depending on the domain, can be chosen independently of the parameter (varepsilon ). Further, we construct an extension operator for second order Sobolev spaces on perforated domains with operator norm independent of (varepsilon ).

In this paper, as a step towards a unified mathematical treatment of the gauge functionals from quantum field theory that have found profound applications in mathematics, we generalize the Seiberg–Witten functional that in particular includes the Kapustin–Witten functional as a special case. We first demonstrate the smoothness of weak solutions to this generalized functional. We then establish the existence of weak solutions under the assumption that the structure group of the bundle is abelian, by verifying the Palais–Smale compactness.

The paper offers a second order necessary condition for a strong minimum in the standard problem of calculus of variations. No idea of such a result seems to have appeared in the classical theory. But a simple example given in the paper shows that the condition can work when all known conditions fail. At the same time, the proof of the proposition is fairly simple. It is also explained in the paper that the condition effectively works only for problems with integrands not convex with respect to the last (derivative) argument.

Let (Gamma ) be a smooth, closed, oriented, ((n-1))-dimensional submanifold of (mathbb {R}^{n+1}). It was shown by Chodosh–Mantoulidis–Schulze [6] that one can perturb (Gamma ) to a nearby (Gamma ') such that all minimizing currents with boundary (Gamma ') are smooth away from a set with Hausdorff dimension less than (n-9). We prove that the perturbation can be made such that the singular set of the minimizing current with boundary (Gamma ') has Minkowski dimension less than (n-9).

Quantitative stability for crystalline anisotropic perimeters, with control on the oscillation of the boundary with respect to the corresponding Wulff shape, is proven for (nge 3). This extends a result of Neumayer (SIAM J Math Anal 48:172–1772, 2016) in (n=2).