Calculus of Variations and Partial Differential Equations最新文献

We look for solutions to the Schrödinger equation

coupled with the mass constraint (int _{mathbb {R}^N}|u|^2,dx = rho ^2), with (Nge 2). The behaviour of g at the origin is allowed to be strongly sublinear, i.e., (lim _{srightarrow 0}g(s)/s = -infty ), which includes the case

with (alpha > 0) and (mu in mathbb {R}), (2 < p le 2^*) properly chosen. We consider a family of approximating problems that can be set in (H^1(mathbb {R}^N)) and the corresponding least-energy solutions, then we prove that such a family of solutions converges to a least-energy one to the original problem. Additionally, under certain assumptions about g that allow us to work in a suitable subspace of (H^1(mathbb {R}^N)), we prove the existence of infinitely, many solutions.

We study the behaviour of global minimizers of a continuum Landau–de Gennes energy functional for nematic liquid crystals, in three-dimensional axially symmetric domains diffeomorphic to a ball (a nematic droplet) and in a restricted class of (mathbb {S}^1)-equivariant configurations. It is known from our previous paper (Dipasquale et al. in J Funct Anal 286:110314, 2024) that, assuming smooth and uniaxial (e.g. homeotropic) boundary conditions and a physically relevant norm constraint in the interior (Lyuksyutov constraint), minimizing configurations are either of torus or of split type. Here, starting from a nematic droplet with the homeotropic boundary condition, we show how singular (split) solutions or smooth (torus) solutions (or even both) for the Euler–Lagrange equations do appear as energy minimizers by suitably deforming either the domain or the boundary data. As a consequence, we derive symmetry breaking results for the minimization among all competitors.

Let (Gamma ) be a compact codimension-two submanifold of ({mathbb {R}}^n), and let L be a nontrivial real line bundle over (X = {mathbb {R}}^n {setminus } Gamma ). We study the Allen–Cahn functional,

on the space of sections u of L. Specifically, we are interested in critical sections for this functional and their relation to minimal hypersurfaces with boundary equal to (Gamma ). We first show that, for a family of critical sections with uniformly bounded energy, in the limit as (varepsilon rightarrow 0), the associated family of energy measures converges to an integer rectifiable ((n-1))-varifold V. Moreover, V is stationary with respect to any variation which leaves (Gamma ) fixed. Away from (Gamma ), this follows from work of Hutchinson–Tonegawa; our result extends their interior theory up to the boundary (Gamma ). Under additional hypotheses, we can say more about V. When V arises as a limit of critical sections with uniformly bounded Morse index, (Sigma := {{,textrm{supp},}}Vert VVert ) is a minimal hypersurface, smooth away from (Gamma ) and a singular set of Hausdorff dimension at most (n-8). If the sections are globally energy minimizing and (n = 3), then (Sigma ) is a smooth surface with boundary, (partial Sigma = Gamma ) (at least if L is chosen correctly), and (Sigma ) has least area among all surfaces with these properties. We thus obtain a new proof (originally suggested in a paper of Fröhlich and Struwe) that the smooth version of Plateau’s problem admits a solution for every boundary curve in ({mathbb {R}}^3). This also works if (4 le nle 7) and (Gamma ) is assumed to lie in a strictly convex hypersurface.

We consider viscosity solutions to two-phase free boundary problems for the p(x)-Laplacian with non-zero right hand side. We prove that flat free boundaries are (C^{1,gamma }). No assumption on the Lipschitz continuity of solutions is made. These regularity results are the first ones in literature for two-phase free boundary problems for the p(x)-Laplacian and also for two-phase problems for singular/degenerate operators with non-zero right hand side. They are new even when (p(x)equiv p), i.e., for the p-Laplacian. The fact that our results hold for merely viscosity solutions allows a wide applicability.

We investigate the existence of solutions to viscous ergodic Mean Field Games systems in bounded domains with Neumann boundary conditions and local, possibly aggregative couplings. In particular we exploit the associated variational structure and search for constrained minimizers of a suitable functional. Depending on the growth of the coupling, we detect the existence of global minimizers in the mass subcritical and critical case, and of local minimizers in the mass supercritical case, notably up to the Sobolev critical case.

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.