Calculus of Variations and Partial Differential Equations最新文献

We consider the regularized Landau-Pekar equations with positive speed of sound and prove the existence of subsonic traveling waves. We provide a definition of the effective mass for the regularized Landau-Pekar equations based on the energy-velocity expansion of subsonic traveling waves. Moreover we show that this definition of the effective mass agrees with the definition based on an energy-momentum expansion of low energy states.

In this paper, we study the Navier–Stokes–Korteweg equations governed by the evolution of compressible fluids with capillarity effects. We first investigate the global well-posedness of solution in the critical Besov space for large initial data. Contrary to pure parabolic methods in Charve et al. (Indiana Univ Math J 70:1903–1944, 2021), we also take the strong dispersion due to large capillarity coefficient (kappa ) into considerations. By establishing a dissipative–dispersive estimate, we are able to obtain uniform estimates and incompressible limits in terms of (kappa ) simultaneously. Secondly, we establish the large time behaviors of the solution. We would make full use of both parabolic mechanics and dispersive structure which implicates our decay results without limitations for upper bound of derivatives while requiring no smallness for initial assumption.

In this paper, we investigate the existence of weak solutions for a class of degenerate elliptic Dirichlet problems with critical nonlinearity and a logarithmic perturbation, i.e.

where ((x,y)in Omega subset mathbb {R}^N = mathbb {R}^m times mathbb {R}^n) with (m ge 1), (nge 0), (Omega cap {x=0}ne emptyset ) is a bounded domain, the parameter (alpha ge 0) and ( Q=m+ n(alpha +1)) denotes the “homogeneous dimension” of (mathbb {R}^N). When (lambda =0), we know that from [23] the problem (0.2) has a Pohožaev-type non-existence result. Then for (lambda in mathbb {R}backslash {0}), we establish the existences of non-negative ground state weak solutions and non-trivial weak solutions subject to certain conditions.

In this paper we prove Gamma-convergence of a nonlocal perimeter of Minkowski type to a local anisotropic perimeter. The nonlocal model describes the regularizing effect of adversarial training in binary classifications. The energy essentially depends on the interaction between two distributions modelling likelihoods for the associated classes. We overcome typical strict regularity assumptions for the distributions by only assuming that they have bounded BV densities. In the natural topology coming from compactness, we prove Gamma-convergence to a weighted perimeter with weight determined by an anisotropic function of the two densities. Despite being local, this sharp interface limit reflects classification stability with respect to adversarial perturbations. We further apply our results to deduce Gamma-convergence of the associated total variations, to study the asymptotics of adversarial training, and to prove Gamma-convergence of graph discretizations for the nonlocal perimeter.

For the Cauchy problem of nonlinear elastic wave equations of three-dimensional isotropic, homogeneous and hyperelastic materials satisfying the null condition, global existence of classical solutions with small initial data was proved in Agemi (Invent Math 142:225–250, 2000) and Sideris (Ann Math 151:849–874, 2000), independently. In this paper, we will consider the asymptotic behavior of global solutions. We first show that the global solution will scatter, i.e., it will converge to some solution of linear elastic wave equations as time tends to infinity, in the energy sense. We also prove the following rigidity result: if the scattering data vanish, then the global solution will also vanish identically. The variational structure of the system will play a key role in our argument.

In this paper, we prove sharp isoperimetric inequalities for lower order eigenvalues of Neumann Laplacian on bounded domains in both compact and noncompact rank-1 symmetric spaces. Our results generalize the work of Wang and Xia for bounded domains in the hyperbolic space (Xia and Wang in Math Ann 385(1–2):863–879, 2023), and Szegö–Weinberger inequalities in rank-1 symmetric spaces obtained by Aithal and Santhanam (Trans Am Math Soc 348(10):3955–3965, 1996).

We consider the following class of elliptic Kirchhoff-Boussinesq type problems given by

where (Omega subset mathbb {R}^{N}) is a bounded and smooth domain, (2< ple frac{2N}{N-2}) for (Nge 3), (2_{**}=frac{2N}{N-4}) if (Nge 5), (2_{**}=infty ) if (3le N <5) and f is a continuous function. We show existence and multiplicity of nontrivial solutions using minimization technique on the Nehari manifold, Mountain Pass Theorem and Genus theory. In this paper we consider the subcritical case (beta =0) and the critical case (beta =1).

We show that the classical Brezis–Nirenberg problem

admits nodal solutions clustering around a point on the boundary of (Omega ) as (varepsilon rightarrow 0), for smooth bounded domains (Omega subset {mathbb {R}^N}) in dimensions (Nge 7).

In this treatise, we discuss existence and uniqueness questions for parametric minimal surfaces in Riemannian spaces, which represent minimal graphs. We choose the Riemannian metric suitably such that the variational solution of the Riemannian Dirichlet integral under Dirichlet boundary conditions possesses a one-to-one projection onto a plane. At first we concentrate our considerations on existence results for boundary value problems. Then we study uniqueness questions for boundary value problems and for complete minimal graphs. Here we establish curvature estimates for minimal graphs on discs, which imply Bernstein-type results.

Consider a pair of smooth, possibly noncompact, properly immersed hypersurfaces moving by mean curvature flow, or, more generally, a pair of weak set flows. We prove that if the ambient space is Euclidean space and if the distance between the two surfaces is initially nonzero, then the surfaces remain disjoint at all subsequent times. We prove the same result when the ambient space is a complete Riemannian manifold of nonzero injectivity radius, provided the curvature tensor (of the ambient space) and all its derivatives are bounded.