Usual Sobolev Research Articles

Diffeomorphism Invariant Minimization of Functionals with Nonuniform Coercivity

We consider the minimization of a functional of the calculus of variations, under assumptions that are diffeomorphism invariant. In particular, a nonuniform coercivity condition needs to be considered. We show that the direct methods of the calculus of variations can be applied in a generalized Sobolev space, which is in turn diffeomorphism invariant. Under a suitable (invariant) assumption, the minima in this larger space belong to a usual Sobolev space and are bounded.

The asymptotic behavior of constant sign and nodal solutions of [formula omitted]-Laplacian problems as [formula omitted] goes to 1

In this paper we study the asymptotic behavior of solutions to the (p,q)-equation −Δpu−Δqu=f(x,u)inΩ,u=0on∂Ω,as p→1+, where N≥2, 1<p<q<1∗≔N/(N−1) and f is a Carathéodory function that grows superlinearly and subcritically. Based on a Nehari manifold treatment, we are able to prove that the (1,q)-Laplace problem given by −div∇u|∇u|−Δqu=f(x,u)inΩ,u=0on∂Ω,has at least two constant sign solutions and one sign-changing solution, whereby the sign-changing solution has least energy among all sign-changing solutions. Furthermore, the solutions belong to the usual Sobolev space W01,q(Ω) which is in contrast with the case of 1-Laplacian problems, where the solutions just belong to the space BV(Ω) of all functions of bounded variation. As far as we know this is the first work dealing with (1,q)-Laplace problems even in the direction of constant sign solutions.

Solving Abel integral equations by regularisation in Hilbert scales

Integral operators of Abel type of order a>0 arise naturally in a large spectrum of physical processes. Their inversion requires care since the resulting inverse problem is ill-posed. The purpose of this work is to devise and analyse a family of appropriate Hilbert scales so that the operator is ill-posed of order a in the scale. We provide weak regularity assumptions on the kernel underlying the operator for the above to hold true. Our construction leads to a well-defined regularisation strategy by Tikhonov regularisation in Hilbert scales. We thereby generalise the results of Gorenflo and Yamamoto for a<1 to arbitrary a>0 and more general kernels. Thanks to tools from interpolation theory, we also show that the a priori information associated to the Hilbert scale formulates in terms of smoothness in usual Sobolev spaces up to boundary conditions, and that the regularisation term actually amounts to penalising derivatives. Finally, following the theoretical construction, we develop a comprehensive numerical approach, where the a priori smoothness is encoded in a single parameter rather than in a full operator. Several numerical examples are shown, both confirming the theoretical convergence rates and showing the general applicability of the method.

Global solutions to the bipolar non-isentropic Euler-Maxwell system in the Besov framework

ABSTRACT This paper investigates the bipolar non-isentropic compressible Euler-Maxwell system in R 3 and T 3 . For both problems, we establish the global existence of smooth solutions in the general Besov spaces, which covers the usual Sobolev spaces with higher regularity and the critical Besov space, when the initial perturbations around the constant states are small enough. As a byproduct, we obtain the large-time asymptotic behavior of the global solutions near the equilibrium state in the general Besov spaces with relatively lower regularity. The proof is based on the technical Fourier frequency-localization method developed through the Littlewood-Paley theory, but some new development and technique are proposed for treating the strong coupling and nonlinearity for the bipolar non-isentropic case.

Approximation results in Sobolev and fractional Sobolev spaces by sampling Kantorovich operators

The present paper deals with the study of the approximation properties of the well-known sampling Kantorovich (SK) operators in “Sobolev-like settings”. More precisely, a convergence theorem in case of functions belonging to the usual Sobolev spaces for the SK operators has been established. In order to get such a result, suitable Strang-Fix type conditions have been required on the kernel functions defining the above sampling type series. As a consequence, certain open problems related to the convergence in variation for the SK operators have been solved. Then, we considered the above operators in a fractional-type setting. It is well-known that, in the literature, several notions of fractional Sobolev spaces are available, such as, the Gagliardo Sobolev spaces (GSs) defined by means of the Gagliardo semi-norm, or the weak Riemann-Liouville Sobolev spaces (wRLSs) defined by the weak (left and right) Riemann-Liouville fractional derivatives and so on. Here, in order to face the above convergence problem, we introduced a new definition of fractional Sobolev spaces, that we called the tight fractional Sobolev spaces (tfSs) and generated as the intersection of the GSs and the symmetric Sobolev spaces (i.e., that given by the intersection of the left and the right wRLSs). In the latter setting, we obtain one of the main results of the paper, that is a convergence theorem for the SK operators with respect to a suitable norm on tfSs.

Normalized multibump solutions to nonlinear Schrödinger equations with steep potential well

We are concerned with the existence of multibump solutions to the nonlinear Schrödinger equation −Δu+λa(x)u+μu=|u|2σuinRN with an L 2-constraint in the L 2-subcritical case σ ∈ (0, 2/N) and the L 2-supercritical case σ ∈ (2/N, 2*/N), where the usual critical Sobolev exponent is 2* = +∞ if N = 1, 2 and 2* = 2N/(N − 2) if N ⩾ 3. Here will arise as a Lagrange multiplier, and has a bottom int a −1(0) composed of ℓ 0 (ℓ 0 ⩾ 1) connected components , where int a −1(0) is the interior of the zero set of a. When ρ is fixed either large in the L 2-subcritical case or small in the L 2-supercritical case, we construct a ℓ-bump (1 ⩽ ℓ ⩽ ℓ 0) positive normalized solution which is localised at ℓ prescribed components for large λ. The asymptotic profile of the solution is also analysed through taking the limit as λ → +∞, and subsequently as ρ → +∞ in the L 2-subcritical case or ρ → 0+ in the L 2-supercritical case. In particular, we find ℓ-bump normalized solutions to the related Dirichlet problem 0\\quad \ ext{for}\\ i=1,\\dots ,\\ell .\\hfill \\end{aligned}\\right.\\end{equation*}?> −Δv+μv=|v|2σv,v∈H01(∪i=1ℓΩi),∑i=1ℓ∫Ωiv2=ρ,v|Ωi>0fori=1,…,ℓ.

Ritz-type projectors with boundary interpolation properties and explicit spline error estimates

In this paper we construct Ritz-type projectors with boundary interpolation properties in finite dimensional subspaces of the usual Sobolev space and we provide a priori error estimates for them. The abstract analysis is exemplified by considering spline spaces and we equip the corresponding error estimates with explicit constants. This complements our results recently obtained for explicit spline error estimates based on the classical Ritz projectors in (Numer Math 144(4):889–929, 2020).

Improved Trudinger–Moser inequality involving Lp-norm on a closed Riemann surface with isometric group actions

In this paper, using the method of blow-up analysis, the authors obtain an improved Trudinger–Moser inequality involving [Formula: see text]-norm ([Formula: see text]) and prove the existence of its extremal function on a closed Riemann surface [Formula: see text] with the action of a finite isometric group [Formula: see text]. To be exact, let [Formula: see text] be the usual Sobolev space, a function space [Formula: see text] and [Formula: see text] [Formula: see text] and [Formula: see text] where [Formula: see text] stands for the number of all distinct points in the set [Formula: see text]. Define [Formula: see text], where [Formula: see text] is the standard [Formula: see text]-norm on [Formula: see text]. Using blow-up analysis, we prove that if [Formula: see text], the supremum [Formula: see text] and this supremum can be attained; if [Formula: see text], the above supremum is infinite. This kind of inequality will play an important role in the study of prescribing Gaussian curvature problem and mean field equations. In particular, their result generalizes those of Chen [A Trudinger inequality on surfaces with conical singularities, Proc. Amer. Math. Soc. 108 (1990) 821–832], Yang [Extremal functions for Trudinger–Moser inequalities of Adimurthi–Druet type in dimension two, J. Differential Equations 258 (2015) 3161–3193] and Fang–Yang [Trudinger–Moser inequalities on a closed Riemannian surface with the action of a finite isometric group, Ann. Sc. Norm. Super. Pisa Cl. Sci. 20 (2020) 1295–1324].

Soliton solutions for a class of Schrödinger equations with a positive quasilinear term and critical growth

AbstractWe consider the following class of quasilinear Schrödinger equations proposed in plasma physics and nonlinear optics $-\Delta u+V(x)u+\frac {\kappa }{2}[\Delta (u^{2})]u=h(u)$ in the whole two-dimensional Euclidean space. We establish the existence and qualitative properties of standing wave solutions for a broader class of nonlinear terms $h(s)$ with the critical exponential growth. We apply the dual approach to obtain solutions in the usual Sobolev space $H^{1}(\mathbb {R}^{2})$ when the parameter $\kappa &gt;0$ is sufficiently small. Minimax techniques, Trudinger–Moser inequality and the Nash–Moser iteration method play an essential role in establishing our results.

Nonlinear Stability of MHD Contact Discontinuities with Surface Tension

We consider the motion of two inviscid, compressible, and electrically conducting fluids separated by an interface across which there is no fluid flow in the presence of surface tension. The magnetic field is supposed to be nowhere tangential to the interface. This leads to the characteristic free boundary problem for contact discontinuities with surface tension in three-dimensional ideal compressible magnetohydrodynamics (MHD). We prove the nonlinear structural stability of MHD contact discontinuities with surface tension in Sobolev spaces by a modified Nash--Moser iteration scheme. The main ingredient of our proof is deriving the resolution and tame estimate of the linearized problem in usual Sobolev spaces of sufficiently large regularity. In particular, for solving the linearized problem, we introduce a suitable regularization that preserves the transport-type structure for the linearized entropy and divergence of the magnetic field.

Dynamics of stochastic retarded Benjamin-Bona-Mahony equations on unbounded channels

&lt;p style='text-indent:20px;'&gt;This article is devoted to the asymptotic behaviour of solutions for stochastic Benjamin-Bona-Mahony (BBM) equations with distributed delay defined on unbounded channels. We first prove the existence, uniqueness and forward compactness of pullback random attractors (PRAs). We then establish the forward asymptotic autonomy of this PRA. Finally, we study the non-delay stability of this PRA. Due to the loss of usual compact Sobolev embeddings on unbounded domains, the forward uniform tail-estimates and forward flattening of solutions are used to prove the forward asymptotic compactness of solutions.&lt;/p&gt;

Singularity preserving spectral collocation method for nonlinear systems of fractional differential equations with the right-sided Caputo fractional derivative

The numerical treatment of fractional differential equations in an accurate way is more difficult to tackle than the standard integer-order counterpart, and occasionally non-specialists are unaware of the specific difficulties. In this paper, we consider nonlinear systems of fractional differential equations involving the right-sided Caputo fractional derivatives of order α∈(0,1). The solutions to these equations have low regularity in the usual Sobolev space even for smooth inputs, requiring regularization techniques to control arbitrary error amplification and to get adequate solutions. Since singularities of the solution usually reflect important features of practical problems, numerical methods preserving the singularities of the solution are preferable. Thus, a singularity preserving regularization method is discussed in detail. The convergence analysis is carried out and the optimal error estimates are obtained.

A Trudinger-Moser inequality with mean value zero on a compact Riemann surface with boundary

In this paper, on a compact Riemann surface $(\Sigma, g)$ with smooth boundary $\partial\Sigma$, we concern a Trudinger-Moser inequality with mean value zero. To be exact, let $\lambda_1(\Sigma)$ denotes the first eigenvalue of the Laplace-Beltrami operator with respect to the zero mean value condition and $\mathcal{ S }= \left\{ u \in W^{1,2} (\Sigma, g) : \|\nabla_g u\|_2^2 \leq 1\right.$ and $\left.\int_\Sigma u \,dv_g = 0 \right \},$ where $W^{1,2}(\Sigma, g)$ is the usual Sobolev space, $\|\cdot\|_2$ denotes the standard $L^2$-norm and $\nabla_{g}$ represent the gradient. By the method of blow-up analysis, we obtain \begin{eqnarray*} \sup_{u \in \mathcal{S}} \int_{\Sigma} e^{ 2\pi u^{2} \left(1+\alpha\|u\|_2^{2}\right) }d v_{g} 0$. Based on the similar work in the Euclidean space, which was accomplished by Lu-Yang \cite{Lu-Yang}, we strengthen the result of Yang \cite{Yang2006IJM}.

Compactness of extremals for critical singular Trudinger-Moser functionals

Let Ω be a smooth bounded domain in R2, and W01,2(Ω) be the usual Sobolev space. Assume 0∈Ω. For any 0<β<1, according to Csato-Roy [7] and Yang-Zhu [21], there exists some function uβ∈W01,2(Ω)∩Cloc1(Ω‾∖{0})∩C0(Ω‾) satisfying∫Ωe4π(1−β)uβ2|x|2βdx=supu∈W01,2(Ω),‖u‖W01,2(Ω)≤1⁡∫Ωe4π(1−β)u2|x|2βdx. In this paper, we concern the compactness of the function family (uβ)0<β<1. It is shown that up to a subsequence, uβ converges to some function u0 in C1(Ω‾) as β tends to 0. Furthermore, u0 is an extremal of the supremumsupu∈W01,2(Ω),‖u‖W01,2(Ω)≤1⁡∫Ωe4πu2dx.The geometric meaning reads as follows: If we write g0=dx12+dx22 and gβ=|x|−2β(dx12+dx22) as the Euclidean and conical metrics on Ω respectively, where x=(x1,x2)∈Ω, then as β→0, the extremal family (uβ)0<β<1 of the singular Trudinger-Moser functionalsJβ(u)=∫Ωe4π(1−β)u2dvgβ is compact under the constraint u∈W01,2(Ω,gβ) and ‖u‖W01,2(Ω,gβ)≤1.

Nonlinear stability and existence of compressible vortex sheets in 2D elastodynamics

The nonlinear stability and local existence of compressible vortex sheets for the two-dimensional isentropic elastic fluid are established in the usual Sobolev spaces. The problem has a characteristic free boundary, and the Kreiss–Lopatinskiĭ condition is satisfied only in a weak form. This paper completes the previous works [6,7] of the first three authors where the weakly linear stability of the rectilinear vortex sheets is proved by means of an upper triangularization technique. Our proof is based on certain higher-order energy estimates and an appropriate modification of the Nash–Moser iteration. In particular, the estimate for the normal derivatives of the characteristic variables can be recovered from that for the linearized divergences and vorticities.

Stability of Multidimensional Thermoelastic Contact Discontinuities

We study the system of nonisentropic thermoelasticity describing the motion of thermoelastic nonconductors of heat in two and three spatial dimensions, where the frame-indifferent constitutive relation generalizes that for compressible neo-Hookean materials. Thermoelastic contact discontinuities are characteristic discontinuities for which the velocity is continuous across the discontinuity interface. Mathematically, this renders a nonlinear multidimensional hyperbolic problem with a characteristic free boundary. We identify a stability condition on the piecewise constant background states and establish the linear stability of thermoelastic contact discontinuities in the sense that the variable coefficient linearized problem satisfies a priori tame estimates in the usual Sobolev spaces under small perturbations. Our tame estimates for the linearized problem do not break down when the strength of thermoelastic contact discontinuities tends to zero. The missing normal derivatives are recovered from the estimates of several quantities relating to physical involutions. In the estimate of tangential derivatives, there is a significant new difficulty, namely the presence of characteristic variables in the boundary conditions. To overcome this difficulty, we explore an intrinsic cancellation effect, which reduces the boundary terms to an instant integral. Then we can absorb the instant integral into the instant tangential energy by means of the interpolation argument and an explicit estimate for the traces on the hyperplane.

Solutions for a class of quasilinear Choquard equations with Hardy–Littlewood–Sobolev critical nonlinearity

In this article, we consider the quasilinear Choquard equation with critical nonlinearity in RN: −ε2Δu+V(x)u−ε2uΔu2=∫RN|u|22μ∗|x−y|μdy|u|22μ∗−2u+h(x,u)u,where 0<μ<N, N≥3, 2μ∗=(2N−μ)∕(N−2) is the critical exponent in the sense of Hardy–Littlewood–Sobolev inequality, ε is a real parameter. By using a change of variables, the quasilinear equations are reduced to a semilinear one, whose associated functionals are well defined in the usual Sobolev space and satisfy the geometric conditions of the mountain pass theorem. Under suitable assumptions on V and h, we investigate the existence and multiplicity of solutions for the above problem by using the mountain pass theorem and index theory. In order to overcome the lack of compactness, we apply the concentration-compactness principle.

A Priori Analysis of an Anisotropic Finite Element Method for Elliptic Equations in Polyhedral Domains

Abstract Consider the Poisson equation in a polyhedral domain with mixed boundary conditions. We establish new regularity results for the solution with possible vertex and edge singularities with interior data in usual Sobolev spaces H σ {H^{\sigma}} with σ ∈ [ 0 , 1 ) {\sigma\in[0,1)} . We propose anisotropic finite element algorithms approximating the singular solution in the optimal convergence rate. In particular, our numerical method involves anisotropic graded meshes with fewer geometric constraints but lacking the maximum angle condition. Optimal convergence on such meshes usually requires the pure Dirichlet boundary condition. Thus, a by-product of our result is to extend the application of these anisotropic meshes to broader practical computations with the price to have “smoother” interior data. Numerical tests validate the theoretical analysis.

Existence of solutions for quasilinear Kirchhoff type problems with critical nonlinearity in $\mathbb {R}^N$

We study the existence of solutions for a class of Kirchhoff type problems with critical growth in $\mathbb {R}^N$:$$-\varepsilon ^2\biggl (a+b\int _{\mathbb {R}^N}|\nabla u|^2\,dx\biggr )\Delta u + V(x)u -\varepsilon ^2a\Delta (u^2)u = |u|^{22^\ast -2}u + h(x,u),$$ $(t, x) \in \mathbb {R} \times \mathbb {R}^N$. By using a change of variables, the quasilinear equations are reduced to a semilinear one, whose associated functionals are well defined in the usual Sobolev space and satisfy the geometric conditions of the mountain pass theorem for suitable assumptions. We prove that it has at least one solution and for any $m \in \mathbb {N}$, it has at least $m$ pairs of solutions. The proofs are based on the variational methods and concentration-compactness principle.

Dynamics of fractional stochastic reaction-diffusion equations on unbounded domains driven by nonlinear noise

This paper is concerned with the asymptotic behavior of the solutions of the fractional reaction-diffusion equations with polynomial drift terms of arbitrary order driven by locally Lipschitz nonlinear diffusion terms defined on Rn. We first prove the well-posedness of the equation based on pathwise uniform estimates as well as uniform estimates on average. We then define a mean random dynamical system via the solution operators and prove the existence and uniqueness of weak pullback mean random attractors. We finally establish the existence of invariant measures when the diffusion terms are globally Lipschitz continuous. The uniform estimates on the tails of solutions are employed to prove the tightness of a family of probability distributions of solutions in order to overcome the non-compactness of usual Sobolev embeddings on unbounded domains.