Calculus of Variations and Partial Differential Equations最新文献

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).

In the article we study the Neumann (p, q)-eigenvalue problems in bounded Hölder (gamma )-singular domains (Omega _{gamma }subset {mathbb {R}}^n). In the case (1<p<infty ) and (1<q<p^{*}_{gamma }) we prove solvability of this eigenvalue problem and existence of the minimizer of the associated variational problem. In addition, we establish some regularity results of the eigenfunctions and some estimates of (p, q)-eigenvalues.

In this paper, we prove BV regularity on the transport density in the mass transport problem to the boundary in two dimensions under certain conditions on the domain, the boundary cost and the mass distribution. Moreover, we show by a counter-example that the smoothness of the mass distribution, the boundary and the boundary cost does not imply that the transport density is (W^{1,p}), for some (p>1).

We study the effective geometric motions of an anisotropic Ginzburg–Landau equation with a small parameter (varepsilon >0) which characterizes the width of the transition layer. For well-prepared initial datum, we show that as (varepsilon ) tends to zero the solutions will develop a sharp interface limit which evolves under mean curvature flow. The bulk limits of the solutions correspond to a vector field ({textbf{u}}(x,t)) which is of unit length on one side of the interface, and is zero on the other side. The proof combines the modulated energy method and weak convergence methods. In particular, by a (boundary) blow-up argument we show that ({textbf{u}}) must be tangent to the sharp interface. Moreover, it solves a geometric evolution equation for the Oseen–Frank model in liquid crystals.

We consider the Cauchy problem of the system of nonlinear Schrödinger equations with derivative nonlinearlity. This system was introduced by Colin and Colin (Differ Int Equ 17:297–330, 2004) as a model of laser-plasma interactions. We study existence of ground state solutions and the global well-posedness of this system by using the variational methods. We also consider the stability of traveling waves for this system. These problems are proposed by Colin–Colin as the open problems. We give a subset of the ground-states set which satisfies the condition of stability. In particular, we prove the stability of the set of traveling waves with small speed for 1-dimension.