Narrow Results

The Cauchy problems for nonlinear complex Ginzburg–Landau equations and nonlinear dissipative wave equations are considered in Sobolev spaces. The relation between the order of the nonlinear terms and the regularity of solutions is considered in terms of the scaling arguments, and the existence of local solutions and small global solutions is shown in Sobolev and Besov spaces.

Let be a smoothly bounded pseudoconvex complex manifold with a family of almost complex structures {ℒ^τ}_{τ ∈ I}, 0 ∈ I. Assume that there is a smooth function λ which is strictly plurisubharmonic with respect to the structure ℒ⁰ in a neighborhood of bM. Then we prove the global regularity for -Neumann problem in Sobolev spaces both in space and parameter variables.

In this paper, we show that the solution map of the generalized Degasperis–Procesi (gDP) equation is not uniformly continuous in Sobolev spaces $H^s(ℝ)$ for $s>3/2$. Our proof is based on the estimates for the actual solutions and the approximate solutions, which consist of a low frequency and a high frequency part. It also exploits the fact that the gDP equation conserves a quantity which is equivalent to the $L^2$ norm.

We discuss compactness of the $\overline{\partial }$-Neumann operator in the setting of weighted $L^2$-spaces on ${ℂ}^n.$ In addition we describe an approach to obtain the compactness estimates for the $\overline{\partial }$-Neumann operator. For this purpose we have to define appropriate weighted Sobolev spaces and prove an appropriate Rellich–Kondrachov lemma.

The purpose of this paper is to explore the geometry of a smooth CR manifold of hypersurface type and its relationship to the higher regularity properties of the complex Green operator on $(0,q)$-forms in the $L^2$-Sobolev space $W_{0,q}^k(M)$ for a fixed $k\in \mathbb{N}$ and $1\le q\le n-2$.

We study Hamiltonian flows in a real separable Hilbert space endowed with a symplectic structure. Measures on the Hilbert space that are invariant with respect to the group of symplectomorphisms preserving two-dimensional symplectic subspaces are investigated. This construction gives the opportunity to present a random Hamiltonian flow in phase space by means of a random unitary group in the space of functions that are quadratically integrable by invariant measure. The properties of mean values of random shift operators are studied.

The question of spectral analysis for deterministic chaos is not well understood in the literature. In this paper, using iterates of chaotic interval maps as time series, we analyze the mathematical properties of the Fourier series of these iterates. The key idea is the connection between the total variation and the topological entropy of the iterates of the interval map, from where special properties of the Fourier coefficients are obtained. Various examples are given to illustrate the applications of the main theorems.

We establish some existence and uniqueness results for a nonlinear elliptic equation. The problem has a diffusion matrix A(x, u) such that A(x, s)ξξ ≥ β(s)|ξ|², with β : (s₀, + ∞) ↦ ℝ a continuous, strictly positive function which goes to infinity when s is near s₀. On the other hand, . Also, the right-hand side f belongs to L¹(Ω). We make use of the concept of renormalized solutions adapted to our problem.

This paper proves some simple inequalities for Sobolev vector fields on nice bounded three-dimensional regions, subject to homogeneous mixed normal and tangential boundary data. The fields just have divergence and curl in L². For the limit cases of prescribed zero normal, respectively zero tangential, data on the whole boundary, the inequalities were proved by Friedrichs who called the result the main inequality of vector analysis. For this mixed case, the optimal constants in the inequality are described, together with the fields for which equality holds. The detailed results depend on a special orthogonal decomposition and the analysis of associated eigenvalue problems.

Due to the lack of regularity of the solutions to the hydrostatic approximation of Navier–Stokes equations, an energy identity cannot be deduced. By including certain nonlinear perturbations to the hydrostatic approximation equations, the solutions to the perturbed problem are smooth enough so that they satisfy the corresponding energy identity. The perturbations considered in this paper are of the monotone class. Three kinds of problems are then studied. To do that, we introduce a functional setting and show in every case that the set of smooth functions with compact support is dense in the space where we search for solutions. When the perturbations are small enough in a certain sense, the solutions of the perturbed problem are close to those of the original one. As a result, this gives a new proof of the existence of solutions to the hydrostatic approximation of Navier–Stokes equations. Finally, this regularization technique has been applied to the analysis of a one-equation hydrostatic turbulence model.

This paper deals with the regularity of an invariant measure μ associated to a class of generalized Ornstein–Uhlenbeck operators. Regularity here means that μ is absolutely continuous with respect to a properly chosen Gaussian reference measure σ on a separable Hilbert space H. Moreover, the square root of its Radon–Nikodym derivative ρ should belong to some directional Sobolev space .

In this paper, we establish some estimates related to the Gaussian densities and to Hermite polynomials in order to obtain an almost sure estimate for each term of the Itô-Wiener expansion of the self-intersection local times of the Brownian motion. In dimension $d\ge 4$ the self-intersection local times of the Brownian motion can be considered as a family of measures on the classical Wiener space. We provide some asymptotics relative to these measures. Finally, we try to estimate the quadratic Wasserstein distance between these measures and the Wiener measure.

Let Ω ⊂ ℝⁿ be a bounded open set. If a sequence f_k : Ω → ℝ^N converges to f in L^∞ in a certain "controlled" manner while bounded in W^1,p (1 < p < + ∞) or BV, we show that f ∈ W^1,p (respectively, f ∈ BV) and ∇f_k → ∇f almost everywhere, where ∇f_k and ∇f are the usual gradients if f_k ∈ W^1,p (respectively, the absolutely continuous part of the gradient measures if f_k ∈ BV). Our main theorem generalizes results for Lipschitz mappings. We show by an example that when p = 1, the limit of a sequence of increasing functions may fail to be in W^1,1 and can even be nowhere C¹.

The purpose of this paper is to relate two notions of Sobolev and BV spaces into metric spaces, due to Korevaar and Schoen on the one hand, and Jost on the other hand. We prove that these two notions coincide and define the same p-energies. We review also other definitions, due to Ambrosio (for BV maps into metric spaces), Reshetnyak and finally to the notion of Newtonian–Sobolev spaces. These last approaches define the same Sobolev (or BV) spaces, but with a different energy, which does not extend the standard Dirichlet energy. We also prove a characterization of Sobolev spaces in the spirit of Bourgain, Brezis and Mironescu in terms of "limit" of the space W^s,p as s → 1, 0 < s < 1, and finally following the approach proposed by Nguyen. We also establish the regularity of traces of maps in W^s,p (0 < s ≤ 1 < sp).

This paper deals with a notion of Sobolev space W^{1, p} introduced by Bourgain, Brezis and Mironescu by means of a seminorm involving local averages of finite differences. This seminorm was subsequently used by Ponce to obtain a Poincaré-type inequality. The main results that we present are a generalization of these two works to a non-Euclidean setting, namely that of Carnot groups. We show that the seminorm expressed in terms of the intrinsic distance is equivalent to the L^p-norm of the intrinsic gradient, and provide a Poincaré-type inequality on Carnot groups by means of a constructive approach which relies on one-dimensional estimates. Self-improving properties are also studied for some cases of interest.

For n ≥ 6, using the Lyapunov–Schmidt reduction method, we describe how to construct (scalar curvature) functions on Sⁿ, so that each of them enables the conformal scalar curvature equation to have an infinite number of positive solutions, which form a blow-up sequence. The prescribed scalar curvature function is shown to have C^{n - 1,β} smoothness. We present the argument in two parts. In this first part, we discuss the uniform cancellation property in the Lyapunov–Schmidt reduction method for the scalar curvature equation. We also explore relation between the Kazdan–Warner condition and the first-order derivatives of the reduced functional, and symmetry in the second-order derivatives of the reduced functional.

By using the Lyapunov–Schmidt reduction method without perturbation, we consider existence results for the conformal scalar curvature on $S^n$ ($n\ge 3$) when the prescribed function (after being projected to ${\mathrm{\text{I}}\mathrm{\text{R}}}^n$) has two close critical points, which have the same value (positive), equal “flatness” (“twin”; flatness $<n-2$), and exhibit maximal behavior in certain directions (“pseudo-peaks”). The proof relies on a balance between the two main contributions to the reduced functional — one from the critical points and the other from the interaction of the two bubbles.

We give a self-contained treatment of the existence of a regular solution to the Dirichlet problem for harmonic maps into a geodesic ball on which the squared distance function from the origin is strictly convex. No curvature assumptions on the target are required. In this route we introduce a new deformation result which permits to glue a suitable Euclidean end to the geodesic ball without violating the convexity property of the distance function from the fixed origin. We also take the occasion to analyze the relationships between different notions of Sobolev maps when the target manifold is covered by a single normal coordinate chart. In particular, we provide full details on the equivalence between the notions of traced Sobolev classes of bounded maps defined intrinsically and in terms of Euclidean isometric embeddings.

In this paper, we obtain improved versions of Stein–Weiss and Caffarelli–Kohn–Nirenberg inequalities, involving Besov norms of negative smoothness. As an application of the former, we derive the existence of extremals of the Stein–Weiss inequality in certain cases, some of which are not contained in the celebrated theorem of Lieb [Sharp constants in the Hardy–Littlewood–Sobolev and related inequalities, Ann. of Math. (2)118(2) (1983) 101–116].

We study compactness of embeddings of Sobolev-type spaces of arbitrary integer order into function spaces on domains in ${ℝ}^n$ with respect to upper Ahlfors regular measures $\nu$, that is, Borel measures whose decay on balls is dominated by a power of their radius. Sobolev-type spaces as well as target spaces considered in this paper are built upon general rearrangement-invariant function norms. Several sufficient conditions for compactness are provided and these conditions are shown to be often also necessary, yielding sharp compactness results. It is noteworthy that the only connection between the measure $\nu$ and the compactness criteria is how fast the measure decays on balls. Applications to Sobolev-type spaces built upon Lorentz–Zygmund norms are also presented.