Sobolev Spaces Research Articles

Existence of solutions for elliptic systems involving the fractional p(x)-Laplacian

In this paper, we consider a class of non-homogeneous fractional (p (.), q (.))−Laplacian systems. By using variational methods and combining with the theory of Lebesgue and fractional Sobolev spaces with variable exponents, we prove the existence of solutions to the system.

Weak solutions to coupled quadratic forward backward stochastic differential equations and Sobolev solutions to their related partial differential equations

We establish existence and uniqueness of a weak (in law sense) solution to coupled forward–backward stochastic differential equations (FBSDEs), with possibly discontinuous diffusion coefficient. We assume that the coefficient of the backward component is of quadratic growth. We first prove the existence of a Sobolev solution for the related partial differential equation (PDE) in the Sobolev space , by using compactness arguments. Next, we use Itô–Krylov's formula to get the existence of weak solution to our considered FBSDE. For the uniqueness part, we first establish the weak uniqueness of the FBSDEs then deduce the uniqueness of its related PDE by using the Feynmann–Kac formula.

Optimal semiclassical regularity of projection operators and strong Weyl law

Projection operators arise naturally as one-particle density operators associated to Slater determinants in fields such as quantum mechanics and the study of determinantal processes. In the context of the semiclassical approximation of quantum mechanics, projection operators can be seen as the analogue of characteristic functions of subsets of the phase space, which are discontinuous functions. We prove that projection operators indeed converge to characteristic functions of the phase space and that in terms of quantum Sobolev spaces, they exhibit the same maximal regularity as characteristic functions. This can be interpreted as a semiclassical asymptotic on the size of commutators in Schatten norms. Our study answers a question raised in Chong et al. [J. Eur. Math. Soc. (unpublished) (2024)] about the possibility of having projection operators as initial data. It also gives a strong convergence result in Sobolev spaces for the Weyl law in phase space.

Horizontal p-Kirchhoff equation on the Heisenberg group

We study the existence and multiplicity of solutions both for the isotropic and the anisotropic horizontal p-Kirchhoff equations on the Heisenberg group Hn, via Krasnoselskii's genus. Finally, an interesting question for the anisotropic embedding in the subelliptic literature is presented and it is conjectured that the exponent p‾⁎:=QnQ∑i=1n1pi−n can be the effective critical exponent in the setting of anisotropic horizontal Sobolev space.

Ground State Solutions for a Non-Local Type Problem in Fractional Orlicz Sobolev Spaces

In this paper, we study the following non-local problem in fractional Orlicz–Sobolev spaces: (−ΔΦ)su+V(x)a(|u|)u=f(x,u), x∈RN, where (−ΔΦ)s(s∈(0,1)) denotes the non-local and maybe non-homogeneous operator, the so-called fractional Φ-Laplacian. Without assuming the Ambrosetti–Rabinowitz type and the Nehari type conditions on the non-linearity f, we obtain the existence of ground state solutions for the above problem with periodic potential function V(x). The proof is based on a variant version of the mountain pass theorem and a Lions’ type result in fractional Orlicz–Sobolev spaces.

Sequences of nodal solutions for critical double phase problems with variable exponents

In this paper, we study a double phase problem with both variable exponents. Such problem has a reaction consisting of a Carathéodory perturbation defined only locally and of a critical term. The presence of the critical term does not permit to use results of the critical point theory for the corresponding energy functional. Consequently, using suitable cut-off functions and truncation techniques we focus on an auxiliary coercive problem on which, differently from our main problem, we can act with variational tools. In this way, we are able to produce a sequence of sign-changing solutions to our main problem converging to 0 in L∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^{\\infty }$$\\end{document} and in the Musielak–Orlicz Sobolev space.

Uniqueness for entropy solution to a class of nonlinear degenerate weighted elliptic p(.)-Laplacian problem with L1-data

In the present paper, we prove the uniqueness of entropy solution to a class of nonlinear degenerate weighted elliptic p(·)-Laplacian problems. These particular problems are delineated by Dirichlet-type boundary conditions and L1 data. We use a priori estimates in the framework of weighted variable exponent Sobolev spaces theory.

On some anisotropic unilateral elliptic problems with measure data

Here, we are concerned with the existence results for some nonlinear unilateral elliptic problem in generalized Sobolev space.

Infinitely many solutions for a p(x)-triharmonic equation with Navier boundary conditions

Abstract In this work, we will study the existence of an infinity of solutions of a Navier problem governed by the p ⁢ ( x ) {p(x)} -triharmonic operator using the theory of Ljusternick–Shrilemann and the theory of the variable exponent Sobolev spaces.

Compact Embeddings for Fractional Super and Sub Harmonic Functions with Radial Symmetry

We prove compactness of the embeddings in Sobolev spaces for fractional super and sub harmonic functions with radial symmetry. The main tool is a pointwise decay for radially symmetric functions belonging to a function space defined by finite homogeneous Sobolev norm together with finite L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2$$\\end{document} norm of the Riesz potentials. As a byproduct we prove also existence of maximizers for the interpolation inequalities in Sobolev spaces for radially symmetric fractional super and sub harmonic functions.

Author Correction: On the concentration-compactness principle for anisotropic variable exponent Sobolev spaces and its applications

Author Correction: On the concentration-compactness principle for anisotropic variable exponent Sobolev spaces and its applications

About Sobolev spaces on fractals: fractal gradians and Laplacians

About Sobolev spaces on fractals: fractal gradians and Laplacians

Inclusion Relations Among Fractional Orlicz-Sobolev Spaces and a Littlewood-Paley Characterization

AbstractEmbeddings among fractional Orlicz-Sobolev spaces with different smoothness are characterized. In particular, besides recovering standard embeddings for classical fractional Sobolev spaces, novel results are derived in borderline situations where the latter fail. For instance, limiting embeddings of Pohozhaev-Trudinger-Yudovich type into exponential spaces are offered. The equivalence of Gagliardo-Slobodeckij norms in fractional Orlicz-Sobolev spaces to norms defined via Littlewood-Paley decompositions, oscillations, or Besov type difference quotients is established as well. This equivalence, of independent interest, is a key tool in the proof of the relevant embeddings. They also rest upon a new optimal inequality for convolutions in Orlicz spaces.

Global Cauchy problem for the complex NLKG and sinh-Gordon equations in super-critical spaces

By introducing a class of new function spaces Bp,qσ,s as the resolution spaces, we study the Cauchy problem for the nonlinear Klein-Gordon (NLKG) and sinh-Gordon equations in all spatial dimensions d⩾1,∂t2u+u−Δu+f(u)=0,(u,∂tu)|t=0=(u0,u1), where f(u)=u1+α or f(u)=sinh⁡u−u. We consider the initial data (u0,u1) in super-critical function spaces Eσ,s×Eσ−1,s for which their norms are defined by‖f‖Eσ,s=‖〈ξ〉σ2s|ξ|fˆ(ξ)‖L2,s<0,σ∈R. Any Sobolev space Hκ can be embedded into Eσ,s, i.e., Hκ⊂Eσ,s for any κ,σ∈R and s<0. We show the global existence and uniqueness of the solutions of NLKG if the initial data belong to some Eσ,s×Eσ−1,s (s<0,σ⩾max⁡(d/2−2/α,1/2),α∈N,α⩾4/d) and their Fourier transforms are supported in the first octant, the smallness conditions on the initial data in Eσ,s×Eσ−1,s are not required for the global solutions. Similar results hold for the sinh-Gordon equation if the spatial dimensions d⩾2.

Global differentiable structures for the Fisher-Rao and Kantorovich-Wasserstein-Otto metrics

We develop a class of non-parametric, Banach-Sobolev manifolds of probability measures that, despite having comparatively weak topologies, support the Fisher-Rao and Kantorovich-Wasserstein-Otto (KWO) Riemannian metrics. The manifolds employ the Kaniadakis κ-deformed logarithms in their charts, and are isomorphic to the (whole) model spaces, W1,λ(μ). These are weighted Sobolev spaces with Lebesgue exponents as small as λ=4, making them suitable for approximations. The KWO metric is analysed through a bundle of Markov semigroups having generators with irregular first-order terms. Strong a-priori estimates are obtained for the associated parabolic equations. Together with a weak Poincaré/Lp-convergence theorem, these are used to establish the Hölder continuity of the “velocity field” representation of tangent vectors—a property inherited by the KWO metric. The manifolds provide a simple framework for the study of problems involving both metrics.

On the existence of capacity solution for a perturbed thermistor problem in anisotropic Orlicz–Sobolev spaces

On the existence of capacity solution for a perturbed thermistor problem in anisotropic Orlicz–Sobolev spaces

Advancements in the Analysis of Sobolev Spaces and Function Spaces on Manifolds: Theoretical Framework and Applications

This research paper delves into Sobolev spaces and function spaces on smooth manifolds, revealing fundamental theorems such as existence, embeddings, and compactness properties. Noteworthy results include the Poincare inequality elucidating function behavior on compact manifolds and compactness properties of Sobolev spaces on Riemannian manifolds. The study establishes trace theorems for functions on the boundary and interpolation results between Sobolev spaces. Isoperimetric inequalities and stability under weak convergence contribute to a holistic understanding of geometric and analytical aspects of Sobolev spaces. The research concludes by exploring invariance under diffeomorphisms and compactness in dual spaces, providing a unified framework for analyzing function spaces on manifolds.

Analysis of Norm-Attainability and Convergence Properties of Orthogonal Polynomials in Weighted Sobolev Spaces

This paper explores norm-attainability of orthogonal polynomials in Sobolev spaces, investigating properties like existence, uniqueness, and convergence. It establishes the convergence of these polynomials in Sobolev spaces, addressing orthogonality preservation and derivative behaviors. Spectral properties, including Sturm-Liouville eigenvalue problems, are analyzed, enhancing the understanding of these polynomials. The study incorporates fundamental concepts like reproducing kernels, Riesz representations, and Bessel’s inequality. Results contribute to the theoretical understanding of orthogonal polynomials, with potential applications in diverse mathematical and computational contexts.

Unconditional convergence of eigenfunction expansions for abstract and elliptic operators

We study the most general class of eigenfunction expansions for abstract normal operators with pure point spectrum in a complex Hilbert space. We find sufficient conditions for such expansions to be unconditionally convergent in spaces with two norms and also estimate the degree of this convergence. Our result essentially generalizes and complements the known theorems of Krein and of Krasnosel'skiĭ and Pustyl'nik. We apply it to normal elliptic pseudodifferential operators on compact boundaryless $C^{\infty }$ -manifolds. We find generic conditions for eigenfunction expansions induced by such operators to converge unconditionally in the Sobolev spaces $W^{\ell }_{p}$ with $p&gt;2$ or in the spaces $C^{\ell }$ (specifically, for the $p$ -th mean or uniform convergence on the manifold). These conditions are sufficient and necessary for the indicated convergence on Sobolev or Hörmander function classes and are given in terms of parameters characterizing these classes. We also find estimates for the degree of the convergence on such function classes. These results are new even for differential operators on the circle and for multiple Fourier series.

Semilinear multi-term fractional in time diffusion with memory

In this study, the initial-boundary value problems to semilinear integro-differential equations with multi-term fractional Caputo derivatives are analyzed. A particular case of these equations models oxygen diffusion through capillaries. Under proper requirements on the given data in the model, the classical and strong solvability of these problems for any finite time interval [0, T] are proved via so-called continuation method. The key point in this approach is finding suitable a priori estimates of a solution in the fractional Hölder and Sobolev spaces.