Calculus of Variations and Partial Differential Equations最新文献

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.

This work addresses several aspects of the dependence on p of the higher eigenvalues (lambda _n) to the Robin problem,

Here, (Omega subset {{mathbb {R}}}^N) is a (C^1) bounded domain, (nu ) is the outer unit normal, (Delta _p u = text {div} (|nabla u|^{p-2}nabla u)) stands for the p-Laplacian operator and (bin L^infty (partial Omega )). Main results concern: (a) the existence of the limits of (lambda _n) as (prightarrow 1), (b) the ‘limit problems’ satisfied by the ‘limit eigenpairs’, (c) the continuous dependence of (lambda _n) on p when (1< p <infty ) and (d) the limit profile of the eigenfunctions as (prightarrow 1). The latter study is performed in the one dimensional and radially symmetric cases. Corresponding properties on the Dirichlet and Neumann eigenvalues are also studied in these two special scenarios.

We revisit Yudovich’s well-posedness result for the 2-dimensional Euler equations for an inviscid incompressible fluid on either a sufficiently regular (not necessarily bounded) open set (Omega subset mathbb {R}^2) or on the torus (Omega =mathbb {T}^2). We construct global-in-time weak solutions with vorticity in (L^1cap L^p_{ul}) and in (L^1cap Y^Theta _{ul}), where (L^p_{ul}) and (Y^Theta _{ul}) are suitable uniformly-localized versions of the Lebesgue space (L^p) and of the Yudovich space (Y^Theta ) respectively, with no condition at infinity for the growth function (Theta ). We also provide an explicit modulus of continuity for the velocity depending on the growth function (Theta ). We prove uniqueness of weak solutions in (L^1cap Y^Theta _{ul}) under the assumption that (Theta ) grows moderately at infinity. In contrast to Yudovich’s energy method, we employ a Lagrangian strategy to show uniqueness. Our entire argument relies on elementary real-variable techniques, with no use of either Sobolev spaces, Calderón–Zygmund theory or Littlewood–Paley decomposition, and actually applies not only to the Biot–Savart law, but also to more general operators whose kernels obey some natural structural assumptions.