Calculus of Variations and Partial Differential Equations最新文献

In this paper, we first prove a rigidity result for a Serrin-type partially overdetermined problem in the half-space, which gives a characterization of capillary spherical caps by the overdetermined problem. In the second part, we prove quantitative stability results for the Serrin-type partially overdetermined problem, as well as capillary almost constant mean curvature hypersurfaces in the half-space.

In this paper, for (d ge 1) and (s in (0,frac{d}{2})), we study the Bianchi–Egnell quotient

where (S_{d,s}) is the best Sobolev constant and ({mathcal {B}}) is the manifold of Sobolev optimizers. By a fine asymptotic analysis, we prove that when (d = 1), there is a neighborhood of ({mathcal {B}}) on which the quotient ({mathcal {Q}}(f)) is larger than the lowest value attainable by sequences converging to ({mathcal {B}}). This behavior is surprising because it is contrary to the situation in dimension (d ge 2) described recently in König (Bull Lond Math Soc 55(4):2070–2075, 2023). This leads us to conjecture that for (d = 1), ({mathcal {Q}}(f)) has no minimizer on (dot{H}^s({mathbb {R}}^d) setminus {mathcal {B}}), which again would be contrary to the situation in (d ge 2). As a complement of the above, we study a family of test functions which interpolates between one and two Talenti bubbles, for every (d ge 1). For (d ge 2), this family yields an alternative proof of the main result of König (Bull Lond Math Soc 55(4):2070–2075, 2023). For (d =1) we make some numerical observations which support the conjecture stated above.

The aggregation equation arises naturally in kinetic theory in the study of granular media, and its interpretation as a 2-Wasserstein gradient flow for the nonlocal interaction energy is well-known. Starting from the spatially homogeneous inelastic Boltzmann equation, a formal Taylor expansion reveals a link between this equation and the aggregation equation with an appropriately chosen interaction potential. Inspired by this formal link and the fact that the associated aggregation equation also dissipates the kinetic energy, we present a novel way of interpreting the aggregation equation as a gradient flow, in the sense of curves of maximal slope, of the kinetic energy, rather than the usual interaction energy, with respect to an appropriately constructed transportation metric on the space of probability measures.

This paper is concerned with a class of fourth-order dispersive wave equations with exponential source term. Firstly, by applying the contraction mapping principle, we establish the local existence and uniqueness of the solution. In the spirit of the variational principle and mountain pass theorem, a natural phase space is precisely divided into three different energy levels. Then we introduce a family of potential wells to derive a threshold of the existence of global solutions and blow up in finite time of solution in both cases with sub-critical and critical initial energy. These results can be used to extend the previous result obtained by Alves and Cavalcanti (Calc. Var. Partial Differ. Equ. 34 (2009) 377–411). Moreover, an explicit sufficient condition for initial data leading to blow up result is established at an arbitrarily positive initial energy level.

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.

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).