In this paper, we study spaces of vector-valued functions with continuous Riemann–Liouville or Caputo fractional derivatives of order \alpha>0 . Despite the absence of a classical product rule for these derivatives, we prove that these spaces form Banach algebras, mirroring the classical case \alpha = 1 . Our work offers a comprehensive comparison with classical Hölder spaces and introduces several new contributions: we provide a sharp characterization of fractional differentiability via Hölder-type regularity; we establish optimal continuous embeddings between fractional spaces of different orders; and we describe in detail the structural differences between the Riemann–Liouville and Caputo frameworks. A cornerstone of our results is a vector-valued extension of the classical Hardy–Littlewood theorem, which establishes new inclusion criteria of Hölder spaces into fractional differentiability spaces. Many of our proofs rely on fine integral representations and delicate regularity estimates that, to the best of our knowledge, are entirely new even in the scalar-valued setting.
We investigate here a finite element approximation for the Lamé system of thermo-electro-elastic materials with ideal contact boundary conditions using a penalty method. We furthermore derive some convergence estimates with respect to both the penalty parameter and the discretization parameter.
Using Leray-Schauder degree, the existence of solutions for quasilinear systems of the form (q(t)Phi(x '))' = f(t, x, x '), with nonlinear boundary conditions of mixed type F(x(0), G(x), H(x ')) = 0, x '(infinity) = 0, is proved for various classes of Phi when F and f satisfy suitable sign conditions.
Let (Omega, Sigma, mu) be a finite complete measure space and X and Y Banach spaces. Let gamma(infinity) be a natural mixed topology on the Lebesgue-Bochner space L-infinity(mu, X). We study absolutely summing operators T : L-infinity(mu, X) -> Y between the locally convex space (L-infinity(mu, X), gamma(infinity)) and the Banach space Y. It is shown that every (gamma(infinity), parallel to.parallel to(Y))-continuous nuclear operator T : L-infinity(mu, X) -> Y is an absolutely summing operator between the locally convex space (L-infinity(mu, X), gamma(infinity)) and the Banach space Y.
Let G be a locally compact group, and let (\Phi,\Psi) be a complementary pair of Young functions satisfying the \Delta_{2} -condition. In this article, we consider PM_{\Psi}(G) the Banach algebra of \Psi -pseudomeasures along with its predual, the Orlicz Figà-Talamanca Herz algebra A_{\Phi}(G) . We prove sufficient conditions for the amenability of G in terms of the norm closed topologically invariant subspaces of PM_{\Psi}(G) . Further, when G is amenable and the Young function \Phi satisfies the MA condition, we establish a one-to-one correspondence between certain topologically invariant subalgebras of PM_{\Psi}(G) and the closed subgroups of G . A similar result is obtained for A_{\Phi}(G) , where we derive a bijection between certain topologically invariant subalgebras of A_{\Phi}(G) and the compact subgroups of G .
In this paper, we generalize the identities related to Steffensen’s inequality using the Taylor formula by introducing a positive weight. For these generalized identities, we obtain bounds by applying the weighted Hermite–Hadamard inequality for a convex or concave function of the form |f^{(n)}|^{q} . Furthermore, we generalize known weighted Hermite–Hadamard-type inequalities by using new identities associated with Steffensen’s inequality.
In this paper, we establish the equivalence of weak and viscosity solutions for a homogeneous problem involving a mixed local and nonlocal elliptic operator in a bounded domain \Omega\subset\mathbb{R}^{N} with Lipschitz boundary. We employ a comparison principle and a priori variational estimates to prove that continuous weak solutions are viscosity solutions and bounded viscosity solutions that vanish outside \Omega are weak solutions. Our results are novel and new for mixed local and nonlocal operators, even for p=2 .
In "Remarks on reproducing kernels of some function spaces" [Function spaces, interpolation theory and related topics, Lund, 2000], J. L. Lions considered a reproducing kernel Hilbert space (RKHS) of harmonic functions on a regular domain with Sobolev traces and obtained a formula that expresses the kernel of this space as an integral on the boundary of some derivatives of the Green function associated with the Laplace operator and the homogeneous Dirichlet boundary condition. This result was simplified and extended later by Englis, Lukkassen, Peetre, and Persson (J. Reine Angew. Math., 2004) to more general elliptic systems of even orders. In particular, they emphasized that the resemblance between Lions' type formula and the Hadamard variational formula only appears when the operator is of order 2. In this paper, we investigate some RKHS of a-harmonic functions, where a in (0,1) refers to a fractional exponent of the Laplace operator. For such fractional-order pseudo-differential operators, the local nonhomogeneous Dirichlet problem can be addressed by means of some a-transmission Sobolev spaces, which were introduced by H & ouml;rmander in the sixties and recently developed by Grubb in a series of papers. We deduce from these works a fractional Poisson formula, which is applied to obtain a Lions' type formula. We observe, in particular, that despite the order of the operator not being 2, this formula resembles the Hadamard variational formula that we prove in the companion paper "Pointwise Hadamard variational formula for the fractional Laplacian [Ann. Fac. Sci. Toulouse Math. (6), 2026]". As a complementary remark, we observe that for a family of RKHS associated with the steady Stokes system, a second-order system, there is also a Lions' type formula for their two-point kernels, which turns out not to be similar to the corresponding Hadamard variation formula.
In this paper, we provide a complete characterization of the weighted Hardy inequalities involving the supremum operator, restricted to the cone of non-increasing sequences, for all positive parameters. We reduce such inequalities to equivalent ones on the cone of non-negative sequences. The latter setting provides a broader framework for analysis and significantly expands the range of proofs that can be established.
This paper deals with an initial-boundary value problem of the dispersive-dissipative wave equation having viscoelastic damping term and a nonlinear source. By applying the Faedo-Galerkin method in conjunction with the contraction mapping principle, we rigorously prove the local existence and uniqueness of weak solutions. Within the potential well theory framework, we categorize the initial data into subcritical, critical, and high initial energy levels. This classification enables us to derive optimal conditions for determining whether the solutions exist globally or exhibit finite-time blow-up behavior. By means of Nakao's lemma and through the refined analysis of calculus inequalities, such as the embedding theorem, an exponential decay estimate of the energy functional is obtained. Additionally, through the innovative construction of auxiliary functionals, we obtain explicit upper and lower bounds for the blow-up time. These bounds provide a quantitative description of how the blow-up dynamics depend on the initial energy levels, thereby offering deeper insights into the long-term behavior of the solutions.
This paper is devoted to the boundary behavior of mappings with bounded and finite distortion, which has been actively studied recently. We consider mappings of domains of the Euclidean space that satisfy the inverse Poletsky inequality with an integrable majorant, are open, and discrete, and not necessarily preserve the boundary of a domain. Under some conditions on the geometry of these domains, it is proved that the specified mappings have a continuous boundary extension. The result is valid even in a more general form, when the majorant mentioned above is integrable over almost all concentric spheres centered at each point. We also have proved some results on equicontinuity of the family of above mappings in the closure of a domain.
This paper is devoted to the normalized solutions of the Choquard equations and the magnetic Choquard equations involving different external potentials. Under the L-2-norm constraint, we apply the variational method and the Gagliardo-Nirenberg-type inequality with the Riesz potential to prove the existence of constraint minimizers of the energy functionals. Particularly, we obtain the compactness of minimizing sequences by establishing the relationship between minimal energies with respect to different mass. We extend and improve the research by Alves and Ji [J. Geom. Anal. 32 (2022), no. 5, article no. 165].
This paper is devoted to the study of a nonlinear system of coupled hyperbolic and parabolic equations. Initially, we establish the well-posedness of our system through the Faedo-Galerkin approach and derive its exponential stability by constructing a suitable Lyapunov functional in the non-degenerate case. Subsequently, we address the general stability in the degenerate case using integral inequalities.
We prove the multiplicity and concentration of normalized solutions of critical biharmonic equations with combined nonlinearities in ℝ^N Δ^2u+V(ε x)u=λu+μ|u|^q-2u+|u|^2^**-2u ℝ^N, ∫_ℝ^N|u|^2dx=c^2, where Δ^2 is the biharmonic operator, N≥5, μ,c>0, ε>0, λ∈ℝ, q∈(2,2+8/N), and 2^**=2N/N-4 is the Sobolev critical exponent. The potential V is a bounded and continuous nonnegative function, satisfying some suitable global conditions. Using minimization techniques and a truncation argument, we show that the number of normalized solutions is not less than the number of global minimum points of V when the parameter ε is sufficiently small. To overcome the loss of compactness of the energy functional due to the critical growth, we apply the concentration-compactness principle. To the best of our knowledge, this study is the first contribution regarding the concentration and multiplicity properties of normalized solutions of critical biharmonic equations with combined nonlinearities in ℝ^N. To some extent, the main results included in this paper complement several recent contributions to the study of biharmonic equations with combined nonlinearities.
We consider a Dirichlet problem driven by a differential operator with unbalanced growth and a reaction exhibiting the combined effects of a parametric singular term and a resonant perturbation. Using a combination of variational tools and critical groups, we show that, for all small values of the parameter, the problem has at least two bounded positive solutions.
This paper investigates a class of problems involving a logarithmic double phase operator with variable exponents and right-hand sides that consist of nonlinearities exhibiting subcritical and superlinear growth. Under very general assumptions, we prove the existence of at least two nontrivial bounded weak solutions for such problems whereby the solutions have opposite energy sign. In addition, we give conditions on the nonlinearity under which the solutions turn out to be nonnegative.
We introduce a notion of weak convergence in arbitrary metric spaces. Metric functionals are key in our analysis: weak convergence of sequences in a given metric space is tested against all metric functionals defined on said space. When restricted to bounded sequences in normed linear spaces, we prove that our notion of weak convergence agrees with the standard one.
For the doubly-degenerate parabolic non-Newtonian thin-film equation u(t )+ div(u(n)divided by del Delta u divided by(p-2)del Delta u) = 0, we derive (local versions) of Bernis estimates of the form integral(Omega)u(n-2p)divided by del u divided by(3p)dx + integral(Omega)u(n-p/2)divided by Delta u divided by(3p/2)dx <= c(n, p, d)integral(Omega)u(n)divided by del Delta u divided by(p)dx for functions u is an element of W-p(2)(Omega) with Neumann boundary condition, where 2 <= p < 19/3 and n lies in a certain range. Here, Omega subset of R-d is a smooth convex domain with d < 3p. A particularly important consequence is the estimate integral(Omega)divided by del Delta(u(n+p/p))divided by(p)dx <= c(n, p, d) integral(Omega)u(n)divided by del Delta u divided by(p)dx. The methods used in this article follow the approach of Grun [Z. Anal. Anwendungen 20 (2001), no. 4, 987-998] for the Newtonian case, while addressing the specific challenges posed by the nonlinear higher-order term divided by del Delta u divided by(p-2)del Delta u and the additional degeneracy. The derived estimates are key to establishing further qualitative results, such as the existence of weak solutions, finite propagation of support, and the appearance of a waiting-time phenomenon.
We represent by {W-lambda,t(alpha)}(t >0) the semigroup generated by-L-lambda(alpha), where L-lambda(alpha) is a Hardy operator on a half space. The operator L-lambda(alpha) includes a fractional Laplacian and it is defined by L-lambda(alpha) = (-Delta)(alpha/2)(d)(R)+ + lambda x(d)(-alpha), alpha is an element of (0, 2], lambda >= 0. We prove that, for every k is an element of N, the rho-variation operator V-rho({t(k) partial derivative(k)(t) W-lambda,t(alpha)}) is bounded on L-p(R-+(d), w) for each 1 < p < infinity and w is an element of A(p)(R-+(d)), with A(p)(R-+(d)) being the Muckenhoupt p-class of weights on R-+(d).
In this paper, the partial regularity of the weak solutions to the magnetohydrodynamics (MHD) system in \mathbb{R}^{4} is studied. In order to tackle the lack of compactness arising in the spatially high-dimensional setting, inspired by Wu [Arch. Rational Mech. Anal. 239 (2021), 1771–1808], we use the defect measures and prove the existence of partially regular weak solutions (satisfying certain local energy inequality) to the 4-dimensional MHD system. As an application, we obtain that the 2-dimensional Hausdorff dimension of singular sets of these weak solutions is finite.
