Calculus of Variations and Partial Differential Equations最新文献

In this paper, we investigate variational problems in (mathbb {R}^2) with the Sobolev norm constraints and with the Dirichlet norm constraints. We focus on property of maximizers of the variational problems. Concerning variational problems with the Sobolev norm constraints, we prove that maximizers are ground state solutions of corresponding elliptic equations, while we exhibit an example of a ground state solution which is not a maximizer of corresponding variational problems. On the other hand, we show that maximizers of maximization problems with the Dirichlet norm constraints and ground state solutions of corresponding elliptic equations are the same functions, up to scaling, under suitable setting.

We present a variational framework for studying the existence and regularity of solutions to elliptic free boundary problems that do not necessarily minimize energy. As applications, we obtain mountain pass solutions of critical and subcritical superlinear free boundary problems, and establish full regularity of the free boundary in dimension (N = 2) and partial regularity in higher dimensions.

In this work we study existence, sign and asymptotic behaviour of solutions for a class of elliptic problems of the integral-differential type under the presence of a parameter. A careful analysis of the influence of the referred parameter on the structure of the set of solutions is made, by considering different reaction terms. Among our main contributions are: (1) a positive answer to Remark 2.4 in Allegretto and Barabanova (Proc R Soc Edinb A 126(3):643–663, 1996); (2) a detailed treatment of the associated eigenvalue problem; (3) The first result involving the existence of a ground-state solution for this class of problems.

We use the Landau-de Gennes energy to describe a particle immersed into nematic liquid crystals with a constant applied magnetic field. We derive a limit energy in a regime where both line and point defects are present, showing quantitatively that the close-to-minimal energy is asymptotically concentrated on lines and surfaces nearby or on the particle. We also discuss regularity of minimizers and optimality conditions for the limit energy.

In this paper we prove a generalisation of Schlenk’s theorem about the existence of contractible periodic Reeb orbits on stable, displaceable hypersurfaces in symplectically aspherical, geometrically bounded, symplectic manifolds, to a forcing result for contractible twisted periodic Reeb orbits. We make use of holomorphic curve techniques for a suitable generalisation of the Rabinowitz action functional in the stable case in order to prove the forcing result. As in Schlenk’s theorem, we derive a lower bound for the displacement energy of the displaceable hypersurface in terms of the action value of such periodic orbits. The main application is a forcing result for noncontractible periodic Reeb orbits on quotients of certain symmetric star-shaped hypersurfaces. In this case, the lower bound for the displacement energy is explicitly given by the difference of the two periods. This theorem can be applied to many physical systems including the Hénon–Heiles Hamiltonian and Stark–Zeeman systems. Further applications include a new proof of the well-known fact that the displacement energy is a relative symplectic capacity on ({mathbb {R}}^{2n}) and that the Hofer metric is indeed a metric.

We revisit the following fractional Schrödinger equation

where (varepsilon >0) is a small parameter, ((-Delta )^s) denotes the fractional Laplacian, (sin (0,1)), (pin (2, 2_s^*)), (2_s^*=frac{2N}{N-2s}), (N>2s), (Vin Cbig ({mathbb {R}}^N, [0, +infty )big )) is a general potential. Under various assumptions on V(x) at infinity, including V(x) decaying with various rate at infinity, we introduce a unified penalization argument and give a complete result on the existence and nonexistence of positive solutions. More precisely, we combine a comparison principle with iteration process to detect an explicit threshold value (p_*), such that the above problem admits positive concentration solutions if (pin (p_*, ,2_s^*)), while it has no positive weak solutions for (pin (2,,p_*)) if (p_*>2), where the threshold (p_*in [2, 2^*_s)) can be characterized explicitly by

Moreover, corresponding to the various decay assumptions of V(x), we obtain the decay properties of the solutions at infinity. Our results reveal some new phenomena on the existence and decays of the solutions to this type of problems.

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.