Calculus of Variations and Partial Differential Equations最新文献

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.

In this manuscript, given a metric tensor on the probability simplex, we define differential operators on the Wasserstein space of probability measures on a graph. This allows us to propose a notion of graph individual noise operator and investigate Hamilton–Jacobi equations on this Wasserstein space. We prove comparison principles for viscosity solutions of such Hamilton–Jacobi equations and show existence of viscosity solutions by Perron’s method. We also discuss a model optimal control problem and show that the value function is the unique viscosity solution of the associated Hamilton–Jacobi–Bellman equation.

Suppose (Omega _0,Omega _1) are two bounded strongly (mathbb {C})-convex domains in (mathbb {C}^n), with (nge 2) and (Omega _1supset overline{Omega _0}). Let (mathcal {R}=Omega _1backslash overline{Omega _0}). We call (mathcal {R}) a (mathbb {C})-convex ring. We will show that for a solution (Phi ) to the homogenous complex Monge–Ampère equation in (mathcal {R}), with (Phi =1) on (partial Omega _1) and (Phi =0) on (partial Omega _0), (sqrt{-1}partial {overline{partial }}Phi ) has rank (n-1) and the level sets of (Phi ) are strongly (mathbb {C})-convex.

We revisit the following nonlinear Schrödinger equation

where (varepsilon >0) is a small parameter, (Nge 2) and (1<p<2^*-1). It is known that the Morse index gives a strong qualitative information on the solutions, such as non-degeneracy, uniqueness, symmetries, singularities as well as classifying solutions. Here we compute the Morse index of positive k-peak solutions to above problem when the critical points of V(x) are non-isolated and degenerate. We also give a specific formula for the Morse index of k-peak solutions when the critical point set of V(x) is a low-dimensional ellipsoid. Our main difficulty comes from the non-uniform degeneracy of potential V(x). Our results generalize Grossi and Servadei’s work (Ann Math Pura Appl 186: 433–453, (2007)) to very degenerate (non-admissible) potentials and show that the structure of potentials highly affects the properties of concentrated solutions.

We construct optimal Hardy weights to subcritical energy functionals h associated with quasilinear Schrödinger operators on infinite graphs. Here, optimality means that the weight w is the largest possible with respect to a partial ordering, and that the corresponding shifted energy functional (h-w) is null-critical. Moreover, we show a decay condition of Hardy weights in terms of their integrability with respect to certain integral weights. As an application of the decay condition, we show that null-criticality implies optimality near infinity. We also briefly discuss an uncertainty-type principle, a Rellich-type inequality and examples.

In this paper, we classify the hypersurfaces of (mathbb {S}^2times mathbb {S}^2) with constant sectional curvature. We prove that the constant sectional curvature can only be (frac{1}{2}). We show that any such hypersurface is a parallel hypersurface of a minimal hypersurface in (mathbb {S}^2times mathbb {S}^2), and we establish a one-to-one correspondence between such minimal hypersurface and the solution to the famous “sinh-Gordon equation” ( left( frac{partial ^2}{partial u^2}+frac{partial ^2}{partial v^2}right) h =-tfrac{1}{sqrt{2}}sinh left( sqrt{2}hright) ).

We obtain a complete description of divergent Palais–Smale sequences for the prescribed Ricci curvature functional on compact homogeneous spaces. As an application, we prove the existence of saddle points on generalized Wallach spaces and several types of generalized flag manifolds. We also describe the image of the Ricci map in some of our examples.