Higher Order Sobolev Spaces Research Articles

Global existence of a pair of coupled Cahn-Hilliard equations with nondegenerate mobility and logarithmic potential

We conducted a mathematical investigation on a system of interconnected Cahn-Hilliard equations featuring a logarithmic potential, nondegenerate mobility, and homogeneous Neumann boundary conditions. This system emerges from a model depicting the phase separation of a binary liquid mixture in a thin film. Assuming certain conditions on the initial data, we successfully established the existence, uniqueness, and stability estimates for the weak solution. Our approach involved initially replacing the logarithmic potential with a smooth counterpart, resulting in the regularization of the original problem (Q) into a regularized problem (Qε). Utilizing the Faedo-Galerkin method and compactness arguments, we demonstrated the existence and uniqueness of a solution for (Qε). Subsequently, by letting ε approach zero, we attained the existence of a solution for the original problem (Q). Additionally, we addressed higher regularity aspects of the weak solutions for both (Q) and (Qε). Employing the standard regularity theory for elliptic problems and introducing additional assumptions regarding the domain's boundary and the initial data, we established that the weak solutions belong to higher-order Sobolev spaces.

Asymptotic stability and sharp decay rates to the linearly stratified Boussinesq equations in horizontally periodic strip domain

We consider an initial boundary value problem of the multi-dimensional Boussinesq equations in the absence of thermal diffusion with velocity damping or velocity diffusion under the stress free boundary condition in horizontally periodic strip domain. We prove the global-in-time existence of classical solutions in high order Sobolev spaces satisfying high order compatibility conditions around the linearly stratified equilibrium, the convergence of the temperature to the asymptotic profile, and sharp decay rates of the velocity field and temperature fluctuation in all intermediate norms based on spectral analysis combined with energy estimates. To the best of our knowledge, our results provide first sharp decay rates for the temperature fluctuation and the vertical velocity to the linearly stratified Boussinesq equations in all intermediate norms.

Positive radially symmetric ground states to a bi-harmonic problem

In this paper, we establish the existence of positive solutions to the semilinear problem for bi-harmonic operator $$ (-\Delta)^2u+u=|u|^{p-1}u, \ \ \text{ in }\ R^n. $$ One key tool is to use the Schwartz symmetrization method, in particular the Riesz inequality to this problem. The other tool is the generalized Ni-Strauss Lemma to radially symmetric functions in higher order Sobolev spaces.

Effective Mass Theorems with Bloch Modes Crossings

We study a Schrödinger equation modeling the dynamics of an electron in a crystal in the asymptotic regime of small wave-length comparable to the characteristic scale of the crystal. Using Floquet Bloch decomposition, we obtain a description of the limit of time averaged energy densities. We make a rather general assumption assuming that the initial data are uniformly bounded in a high order Sobolev spaces and that the crossings between Bloch modes are at worst conical. We show that despite the singularity they create, conical crossing do not trap the energy and do not prevent dispersion. We also investigate the interactions between modes that can occurred when there are some degenerate crossings between Bloch bands.

A sharpened form of Adams-type inequalities on higher-order Sobolev spaces : a simple approach

AbstractIn this paper, we develop an extremely simple method to establish the sharpened Adams-type inequalities on higher-order Sobolev spaces $W^{m,\frac {n}{m}}(\mathbb {R}^n)$ in the entire space $\mathbb {R}^n$ , which can be stated as follows: Given $\Phi \left ( t\right ) =e^{t}-\underset {j=0}{\overset {n-2}{\sum }} \frac {t^{j}}{j!}$ and the Adams sharp constant $\beta _{n,m}$ . Then, $$ \begin{align*}\sup_{\|\nabla^mu\|_{\frac{n}{m}}^{\frac{n}{m}}+\|u\|_{\frac{n}{m}}^{\frac{n}{m}}\leq1}\int_{\mathbb{R}^n}\Phi\Big(\beta_{n,m} (1+\alpha \|u\|_{\frac{n}{m}}^{\frac{n}{m}} )^{\frac{m}{n-m}}|u|^{\frac{n}{n-m}}\Big)dx<\infty, \end{align*} $$ for any $0<\alpha <1$ . Furthermore, we construct a proper test function sequence to derive the sharpness of the exponent $\alpha $ of the above Adams inequalities. Namely, we will show that if $\alpha \ge 1$ , then the above supremum is infinite.Our argument avoids applying the complicated blow-up analysis often used in the literature to deal with such sharpened inequalities.

On the regularization effect of stochastic gradient descent applied to least-squares

We study the behavior of stochastic gradient descent applied to $\|Ax -b \|_2^2 \rightarrow \min$ for invertible $A \in \mathbb{R}^{n \times n}$. We show that there is an explicit constant $c_{A}$ depending (mildly) on $A$ such that $$ \mathbb{E} ~\left\| Ax_{k+1}-b\right\|^2_{2} \leq \left(1 + \frac{c_{A}}{\|A\|_F^2}\right) \left\|A x_k -b \right\|^2_{2} - \frac{2}{\|A\|_F^2} \left\|A^T A (x_k - x)\right\|^2_{2}.$$ This is a curious inequality: the last term has one more matrix applied to the residual $u_k - u$ than the remaining terms: if $x_k - x$ is mainly comprised of large singular vectors, stochastic gradient descent leads to a quick regularization. For symmetric matrices, this inequality has an extension to higher-order Sobolev spaces. This explains a (known) regularization phenomenon: an energy cascade from large singular values to small singular values smoothes.

Approximation in higher-order Sobolev spaces and Hodge systems

Let d≥2 be an integer, 1≤l≤d−1 and φ be a differential l-form on Rd with W˙1,d coefficients. It was proved by Bourgain and Brezis ([5, Theorem 5]) that there exists a differential l-form ψ on Rd with coefficients in L∞∩W˙1,d such that dφ=dψ. In the same work, Bourgain and Brezis also left as an open problem the extension of this result to the case of differential forms with coefficients in the higher order space W˙2,d/2 or more generally in the fractional Sobolev spaces W˙s,p with sp=d. We give a positive answer to this question, provided that d−κ≤l≤d−1, where κ is the largest positive integer such that κ<min⁡(p,d). The proof relies on an approximation result (interesting in its own right) for functions in W˙s,p by functions in W˙s,p∩L∞, even though W˙s,p does not embed into L∞ in this critical case. The proofs rely on some techniques due to Bourgain and Brezis but the context of higher order and/or fractional Sobolev spaces creates various difficulties and requires new ideas and methods.

The Sine-Gordon regime of the Landau–Lifshitz equation with a strong easy-plane anisotropy

It is well-known that the dynamics of biaxial ferromagnets with a strong easy-plane anisotropy is essentially governed by the Sine-Gordon equation. In this paper, we provide a rigorous justification to this observation. More precisely, we show the convergence of the solutions to the Landau–Lifshitz equation for biaxial ferromagnets towards the solutions to the Sine-Gordon equation in the regime of a strong easy-plane anisotropy. Moreover, we establish the sharpness of our convergence result.This result holds for solutions to the Landau–Lifshitz equation in high order Sobolev spaces. We first provide an alternative proof for local well-posedness in this setting by introducing high order energy quantities with better symmetrization properties. We then derive the convergence from the consistency of the Landau–Lifshitz equation with the Sine-Gordon equation by using well-tailored energy estimates. As a by-product, we also obtain a further derivation of the free wave regime of the Landau–Lifshitz equation.

Equivalence of critical and subcritical sharp Trudinger–Moser–Adams inequalities

Sharp Trudinger–Moser inequalities on the first order Sobolev spaces and their analogous Adams inequalities on high order Sobolev spaces play an important role in geometric analysis, partial differential equations and other branches of modern mathematics. Such geometric inequalities have been studied extensively by many authors in recent years and there is a vast literature. There are two types of such optimal inequalities: critical and subcritical sharp inequalities, both are with best constants. Critical sharp inequalities are under the restriction of the full Sobolev norms for the functions under consideration, while the subcritical inequalities are under the restriction of the partial Sobolev norms for the functions under consideration. There are subtle differences between these two type of inequalities. Surprisingly, we prove in this paper that these critical and subcritical Trudinger–Moser and Adams inequalities are actually equivalent. Moreover, we also establish the asymptotic behavior of the supremum for the subcritical Trudinger–Moser and Adams inequalities on the entire Euclidean spaces, and provide a precise relationship between the suprema for the critical and subcritical Trudinger–Moser and Adams inequalities. This relationship of supremum is useful in establishing the existence and nonexistence of extremal functions for the Trudinger–Moser inequalities. Since the critical Trudinger–Moser and Adams inequalities can be easier to prove than subcritical ones in some occasions, and more difficult to establish in other occasions, our results and the method suggest a new approach to both the critical and subcritical Trudinger–Moser and Adams type inequalities.

A Class of High Order Nonlocal Operators

We study a class of nonlocal operators that may be seen as high order generalizations of the well known nonlocal diffusion operators. We present properties of the associated nonlocal functionals and nonlocal function spaces including nonlocal versions of Sobolev inequalities such as the nonlocal Poincare and nonlocal Gagliardo–Nirenberg inequalities. Nonlocal characterizations of high order Sobolev spaces in the spirit of Bourgain–Brezis–Mironescu are provided. Applications of nonlocal calculus of variations to the well-posedness of linear nonlocal models of elastic beams and plates are also considered.

Characterizations of Sobolev spaces via averages on balls

In this paper, the authors prove several equivalent characterizations of Sobolev spaces of even integer orders on Rn, using the average Btf(x)≔1|B(x,t)|∫B(x,t)f(y)dy of a function f over the ball B(x,t)≔{y∈Rn:|y−x|<t} with x∈Rn and t∈(0,∞). These characterizations rely only on the metric and the Lebesgue measure on Rn and are simpler than those obtained recently by Alabern et al. (2012). Moreover, these results may shed new light on the theory of high order Sobolev spaces on spaces of homogeneous type.

Superposition operators between Sobolev spaces and a non-existence result of higher-order regular solutions for the [formula omitted]-Laplacian

In this paper, a well-known non-existence result of Pohožaev for a Dirichlet problem with Laplacian is generalized to a non-existence result of higher-order regular strong solutions for a Dirichlet problem with p-Laplacian subjected to the following natural principle: the stronger (respectively weaker) are the assumptions on the given data, the larger (respectively smaller) is the Sobolev space in which no nontrivial solutions can be found. To do this, we use some recent developments on superposition operators between higher-order Sobolev spaces.

Characterization of polynomials and higher-order Sobolev spaces in terms of functionals involving difference quotients

The aim of this paper, which deals with a class of singular functionals involving difference quotients, is twofold: deriving suitable integral conditions under which a measurable function is polynomial and stating necessary and sufficient criteria for an integrable function to belong to a kth-order Sobolev space. One of the main theorems is a new characterization of Wk,p(Ω), k∈N and p∈(1,+∞), for arbitrary open sets Ω⊂Rn. In particular, we provide natural generalizations of the results regarding Sobolev spaces summarized in Brézis’ overview article [Brézis (2002)] to the higher-order case, and extend the work [Borghol (2007)] to a more general setting.

Stability of solitary waves for the vector nonlinear Schrödinger equation in higher-order Sobolev spaces

In this manuscript, a sharp form of orbital stability for the Schrödinger system, also known as the vector NLS, i∂∂tuj+∂2∂xxuj+2∑i=1m|ui|2uj=0, where uj are complex-valued functions of (x,t)∈R2, j=1,2,…,m, is established in L2-based Sobolev classes of arbitrarily high order. The result means practically that not only does the bulk of what emanates from the perturbed solitary wave stay close in shape and propagation and phase speeds to the original solitary wave, but emerging residual oscillations must also be very small and not only in the energy norm.

How to Recognize Polynomials in Higher Order Sobolev Spaces

This paper extends characterizations of Sobolev spaces by Bourgain, Brézis, and Mironescu to the higher order case. As a byproduct, we obtain an integral condition for the Taylor remainder term, which implies that the function is a polynomial. Similar questions are also considered in the context of Whitney jets.

A new approach to sharp Moser–Trudinger and Adams type inequalities: A rearrangement-free argument

The main purpose of this paper is two-fold. On the one hand, we will develop a new approach to establish sharp singular Moser–Trudinger and Adams type inequalities in unbounded domains of Euclidean spaces without using the standard symmetrization. On the other hand, we will prove the sharp singular Adams type inequality on high order Sobolev spaces Wm,nm(Rn) of arbitrary integer order m (Theorem 1.1) which improves the results of Ruf and Sani (2013) [48] where sharp Adams inequalities were established for even m and those of the authors (Lam and Lu, 2012 [28,29]) for odd m but with different and more restricted norms. We first establish the sharp local singular Adams inequality on domains Ω in Rn of finite measure (Theorem 1.4). We take a perspective that any function in the high order Sobolev spaces Wm,nm(Rn) can be represented as a Bessel potential. Thus, we can fully use the tools from harmonic analysis and the kernel properties of the polyharmonic operators (τI−Δ)m2. Once we have established this sharp local Adams inequality, then we can adapt the rearrangement-free method we will develop in this paper to derive a global sharp Adams inequality from a local one. Our argument substantially simplifies those in Ruf and Sani (2013) [48] and Lam and Lu (2012) [28,29] and avoids the use of rather deep and complicated comparison principle of solutions to polyharmonic operators used in Ruf and Sani (2013) [48], Lam and Lu (2012) [28,29]. Moreover, our theorem holds on Sobolev spaces Wα,nα(Rn) of any positive fractional order α<n (Theorems 1.6 and 1.7). As consequences, we can also establish sharp Adams inequalities with less restrictions on the Sobolev norms (see Theorems 1.2, 1.3 and 1.5). Our approach is surprisingly simple and general and can be easily applied to scenarios such as Riemannian and sub-Riemannian manifolds where symmetrization argument does not work (see, e.g., on the Heisenberg group Lam and Lu (2012) [30]).

Intermediate long wave systems for internal waves

This paper, which deals with internal waves in the two-layer formulation, consists of three parts. The first part is devoted to the derivation of asymptotic models in the intermediate long wave (ILW) regime for internal waves with a free upper surface and with a flat bottom. Using a method similar to that introduced by Bona et al (2008 J. Math. Pures Appl. 89 538–66), we obtain a one-parameter ILW system which is consistent with the full Euler system. In the second part we investigate the well-posedness of the ILW system under the rigid lid assumption. The local well-posedness on the time scale in lower order Sobolev spaces and the large time well-posedness on the long time scale in higher order Sobolev spaces are proven for both 1D and 2D cases. The long time existence result seems to be the first of this type in the context of internal waves. In the third part we provide a rigorous justification to show that the Benjamin–Ono (BO) system can be deduced from the ILW system by letting the depth of the lower flow tend to ∞. The convergence results between the solutions of the BO system and the ILW system are established in both the short time scale and the long time scale , in different functional settings.

Sharp singular Adams inequalities in high order Sobolev spaces

In this paper, we prove a version of weighted inequalities of exponential type for fractional integrals with sharp constants in any domain of finite measure in $\mathbb{R}^{n}$. Using this we prove a sharp singular Adams inequality in high order Sobolev spaces in bounded domain at critical case. Then we prove sharp singular Adams inequalities for high order derivatives on unbounded domains. Our results extend the singular Moser-Trudinger inequalities of first order in \cite{Ad2, R, LR, AdY} to the higher order Sobolev spaces $W^{m,\frac{n}{m}}$ and the results of \cite{RS} on Adams type inequalities in unbounded domains to singular case. Our singular Adams inequality on $W^{2,2}(\mathbb{R}^{4})$ with standard Sobolev norm at the critical case settles a unsolved question remained in \cite{Y}.

Selfadjointness of degenerate elliptic operators on higher order Sobolev spaces

Let us consider the operator $A_n u$:=$(-1)^{n+1} \alpha (x) u^{(2n)}$ on $H^n_0(0,1)$ with domain $D(A_n)$:=$\{u\in H^n_0(0,1)\cap H^{2n}$loc$(0,1) : A_n u\in H^n_0(0,1)\}$, where $n\in\N$, $\alpha\in H^n_0(0,1)$, $\alpha (x)>0$ in $(0,1).$ Under additional boundedness and integrability conditions on $\alpha$ with respect to $x^{2n} (1-x)^{2n},$ we prove that $(A_n,D(A_n))$ is nonpositive and selfadjoint, thus it generates a cosine function, hence an analytic semigroup in the right half plane on $H^n_0(0,1)$. Analyticity results are also proved in $H^n (0,1).$ In particular, all results work well when $\alpha (x)=x^{j} (1-x)^{j}$ for $|j-n|<1/2$. Hardy type inequalities are also obtained.

On weighted critical imbeddings of Sobolev spaces

Our concern in this paper lies with two aspects of weighted exponential spaces connected with their role of target spaces for critical imbeddings of Sobolev spaces. We characterize weights which do not change an exponential space up to equivalence of norms. Specifically, we first prove that $${L_{\exp t^{\alpha}}(\chi_B)=L_{\exp t^{\alpha}}(\rho)}$$ if and only if $${\rho^q \in L_q}$$ with some q > 1. Second, we consider the Sobolev space $${W^{1}_{N}(\varOmega),}$$ where Ω is a bounded domain in $${\mathbb{R}^{N}}$$ with a sufficiently smooth boundary, and its imbedding into a weighted exponential Orlicz space $${L_{\exp t^{p'}}(\varOmega,\rho)}$$ , where ρ is a radial and non-increasing weight function. We show that there exists no effective weighted improvement of the standard target $${L_{\exp t^{N'}}(\varOmega)=L_{\exp t^{N'}}(\varOmega,\chi_{\varOmega})}$$ in the sense that if $${W^{1}_{N}(\varOmega)}$$ is imbedded into $${L_{\exp t^{p'}}(\varOmega,\rho),}$$ then $${L_{\exp t^{p'}}(\varOmega,\rho)}$$ and $${L_{\exp t^{N'}}(\varOmega)}$$ coincide up to equivalence of the norms; that is, we show that there exists no effective improvement of the standard target space. The same holds for critical cases of higher-order Sobolev spaces and even Besov and Lizorkin–Triebel spaces.