Band 25, Heft 1

In this paper, we extend our convexity theory for C 2 cost functions in optimal transportation to more general generating functions, which were originally introduced by the second author to extend the framework of optimal transportation to embrace near field geometric optics. In particular we provide an alternative geometric treatment to the previous analytic approach using differential inequalities, which also gives a different derivation of the invariance of the fundamental regularity conditions under duality. We also extend our local theory to cover the strict version of these conditions for C 2 cost and generating functions.

Let V be a finite tree with radially decaying weights. We show that there exists a set E ⊂ R 2 $E\subset {\mathbb{R}}^{2}$ for which the following two problems are equivalent: (1) Given a (real-valued) function ϕ on the leaves of V , extend it to a function Φ on all of V so that ‖ Φ ‖ L 1 , p ( V ) ${\Vert}{\Phi}{{\Vert}}_{{L}^{1,p}\left(V\right)}$ has optimal order of magnitude. Here, L 1, p ( V ) is a weighted Sobolev space on V . (2) Given a function f : E → R $f:E\to \mathbb{R}$ , extend it to a function F ∈ L 2 , p ( R 2 ) $F\in {L}^{2,p}\left({\mathbb{R}}^{2}\right)$ so that ‖ F ‖ L 2 , p ( R 2 ) ${\Vert}F{{\Vert}}_{{L}^{2,p}\left({\mathbb{R}}^{2}\right)}$ has optimal order of magnitude.

In this paper, we mainly extend the classical Adams’ inequality to its anisotropic type. By using the rearrangement argument, we establish best constants for anisotropic Adams’ type inequality with exact growth in R 4 ${\mathbb{R}}^{4}$ . We give the optimal growth rate of the exponential-type function in the whole space R 4 ${\mathbb{R}}^{4}$ . Furthermore, using the same method, we also prove anisotropic Adams’ type inequality on bounded domain Ω ⊂ R 4 ${\Omega}\subset {\mathbb{R}}^{4}$ .

In this paper, we are concerned with n -component Ginzburg-Landau equations on R 2 ${\mathbb{R}}^{2}$ . By introducing a diffusion constant for each component, we discuss that the n -component equations are different from n -copies of the single Ginzburg-Landau equations. Then, the results of Brezis-Merle-Riviere for the single Ginzburg-Landau equation can be nontrivially extended to the multi-component case. First, we show that if the solutions have their gradients in L 2 space, they are trivial solutions. Second, we prove that if the potential is square summable, then it has quantized integrals, i.e., there exists one-to-one correspondence between the possible values of the potential energy and N n ${\mathbb{N}}^{n}$ . Third, we show that different diffusion coefficients in the system are important to obtain nontrivial solutions of n -component equations.

For elliptic partial differential equations, the pure interior C 2 estimates and Pogorelov type estimates are important issues. In this paper, we study the interior estimates of Γ k ̃ $\tilde {{{\Gamma}}_{k}}$ -admissible solutions for curvature quotient equations σ k ( η ( κ ) ) σ l ( η ( κ ) ) 1 k − l = g ( x , u ) ${\left(\frac{{\sigma }_{k}\left(\eta \left(\kappa \right)\right)}{{\sigma }_{l}\left(\eta \left(\kappa \right)\right)}\right)}^{\frac{1}{k-l}}=g\left(x,u\right)$ , and establish the pure interior curvature estimate when l = k − 1, 0 < k ≤ n and Pogorelov type estimate when 0 ≤ l < k ≤ n .

We are concerned with the following nonlinear Hartree system − Δ u + P 1 ( | x | ) u = α 1 | x | − 1 ∗ u 2 u + β | x | − 1 ∗ v 2 u   in R 3 , − Δ v + P 2 ( | x | ) v = α 2 | x | − 1 ∗ v 2 v + β | x | − 1 ∗ u 2 v   in R 3 , $$\begin{cases}-{\Delta}u+{P}_{1}\left(\vert x\vert \right)u={\alpha }_{1}\left(\vert x{\vert }^{-1}\ast {u}^{2}\right)u+\beta \left(\vert x{\vert }^{-1}\ast {v}^{2}\right)u\quad \hfill & \text{in} {\mathbb{R}}^{3},\hfill \\ -{\Delta}v+{P}_{2}\left(\vert x\vert \right)v={\alpha }_{2}\left(\vert x{\vert }^{-1}\ast {v}^{2}\right)v+\beta \left(\vert x{\vert }^{-1}\ast {u}^{2}\right)v\quad \hfill & \text{in} {\mathbb{R}}^{3},\hfill \end{cases}$$ where P 1 ( r ) and P 2 ( r ) are positive radial potentials, α 1 > 0, α 2 > 0 and β ∈ R $\beta \in \mathbb{R}$ is a coupling constant. We first study nondegeneracy of ground states for the limit system of the above problem. As applications, we show that the nonlinear Hartree system has infinitely many non-radial positive synchronized solutions, whose energy can be arbitrarily large.

We consider the Boussinesq equation on the line for a broad class of Schwartz initial data relevant for water waves. In a recent work, we identified ten main sectors describing the asymptotic behavior of the solution, and for each of these sectors we gave an exact expression for the leading asymptotic term in the case when no solitons are present. In this paper, we derive an asymptotic formula in Sector IV, characterized by x t ∈ ( 1 3 , 1 ) $\frac{x}{t}\in \left(\frac{1}{\sqrt{3}},1\right)$ , in the case when solitons are present. In particular, our results provide an exact expression for the soliton-radiation interaction to leading order and a verification of the soliton resolution conjecture for the Boussinesq equation in Sector IV.

The purpose of this paper is two-fold. First, we derive sharp Trudinger–Moser inequalities with logarithmic weights in fractional dimensions:   sup ∫ 0 1 w ( r ) u ′ ( r ) β + 2 d λ α 1 / ( β + 2 ) ≤ 1 ∫ 0 1 e μ α , θ , γ u β + 2 β + 1 1 − γ d λ θ < + ∞ , $$\,\underset{{\left(\underset{0}{\overset{1}{\int }}w\left(r\right){\left\vert {u}^{\prime }\left(r\right)\right\vert }^{\beta +2}\mathrm{d}{\lambda }_{\alpha }\right)}^{1/\left(\beta +2\right)}\le 1}{\mathrm{sup}}\underset{0}{\overset{1}{\int }}{\text{e}}^{{\mu }_{\alpha ,\theta ,\gamma }{\left\vert u\right\vert }^{\frac{\beta +2}{\left(\beta +1\right)\left(1-\gamma \right)}}}\mathrm{d}{\lambda }_{\theta }{< }+\infty ,$$ where 0 ≤ γ < 1, α = β + 1, μ α , θ , γ ≔ θ + 1 ω α 1 / α 1 − γ 1 1 − γ ${\mu }_{\alpha ,\theta ,\gamma }{:=}\left(\theta +1\right){\left[{\omega }_{\alpha }^{1/\alpha }\left(1-\gamma \right)\right]}^{\frac{1}{1-\gamma }}$ , w ( r ) = w 1 ( r ) = log 1 r γ β + 1 $w\left(r\right)={w}_{1}\left(r\right)={\left(\mathrm{log}\frac{1}{r}\right)}^{\gamma \left(\beta +1\right)}$ or w ( r ) = w 2 ( r ) = log e r γ β + 1 $w\left(r\right)={w}_{2}\left(r\right)={\left(\mathrm{log}\frac{e}{r}\right)}^{\gamma \left(\beta +1\right)}$ and λ θ ( E ) = ω θ ∫ E r θ d r for all E ⊂ R $E\subset \mathbb{R}$ . The case γ > 1 and γ = 1 are also be considered in this part to improve our paper. Indeed, we have a continuous embedding X ( w 2 ) ↪ L ∞ (0, 1) for γ > 1 and a critical growth of double exponential type for γ = 1. Second, we apply the Lions type Concentration-Compactness principle for Trudinger–Moser inequalities and the precise estimate of normalized concentration limit for normalized concentrating sequence at origin to establish the existence of extremals for Trudinger–Moser inequalities when w ( r ) = w 1 ( r ) = log 1 r γ β + 1 $w\left(r\right)={w}_{1}\left(r\right)={\left(\mathrm{log}\frac{1}{r}\right)}^{\gamma \left(\beta +1\right)}$ and γ > 0 is sufficiently small.

We study the existence problem for semilinear equations ( E ): − Au = f (⋅, u ) + μ , with Borel measure μ and operator A that generates a symmetric Markov semigroup. We merely assume that the nonlinear part f is a Carathéodory function satisfying the so-called sign condition. We extend the method of sub and supersolutions for ( E ) and prove that if such exist, then there exists a solution to ( E ) (we do not even assume that the subsolution is less than or equal to the supersolution!). We further show that for any μ there exists a unique metric projection μ ̂ $\hat{\mu }$ of μ onto the set of good measures, i.e. Borel measures for which there exists a solution to ( E ).

We consider incompressible inviscid elastodynamics equations with a free surface and establish regularity of solutions for these equations. Compared with previous result on this free boundary problem [X. Gu and F. Wang, Well-posedness of the free boundary problem in incompressible elastodynamic under the mixed type stability condition, J. Math. Anal. Appl., 482, (2020), 123529] in space H 3 , we are able to establish regularity in space H 2.5+ δ upon the Arbitrary Lagrangian-Eulerian (ALE) coordinates. It is achieved by reformulating the system into a new formulation with the ALE coordinates, presenting uniform estimates for the pressure, tangential estimates for the system, as well as curl and divergence estimates.

This paper is concerned with the existence of normalized solutions to a class of (2, q )-Laplacian equations in all the possible cases with respect to the mass critical exponents 2(1 + 2/ N ), q (1 + 2/ N ). In the mass subcritical cases, we study a global minimization problem and obtain a ground state solution. While in the mass critical cases, we prove several nonexistence results. At last, we derive a ground state and infinitely many radial solutions in the mass supercritical case. Compared with the classical Schrödinger equation, the (2, q )-Laplacian equation possesses a quasi-linear term, which brings in some new difficulties and requires a more subtle analysis technique. Moreover, the vector field a ⃗ ( ξ ) = | ξ | q − 2 ξ $ \overrightarrow {a}\left(\xi \right)=\vert \xi {\vert }^{q-2}\xi $ corresponding to the q -Laplacian is not strictly monotone when q < 2, so we shall consider separately the case q < 2 and the case q > 2.