This paper investigates the [Formula: see text]-stability and [Formula: see text] control design for a class of systems characterized by mixed time-varying delays (MTVDs). A novel analysis framework is developed based on a Lyapunov–Krasovskii functional (LKF), tailored to capture the dynamics of such systems more effectively than conventional direct methods. A set of sufficient conditions expressed in terms of linear matrix inequalities (LMIs) is derived to guarantee [Formula: see text]-stability and achieve a prescribed [Formula: see text] performance bound [Formula: see text] in the absence of external inputs. Furthermore, a state-feedback control strategy is formulated to ensure the closed-loop system maintains [Formula: see text]-stability while satisfying the [Formula: see text] performance criterion. The proposed method is validated through numerical case studies involving various forms of the function [Formula: see text], illustrating the practical applicability and effectiveness of the theoretical results.
A bidouble cover between complex algebraic varieties is a flat [Formula: see text]-Galois cover [Formula: see text]. In this situation, there exist three intermediate quotients [Formula: see text] and [Formula: see text] which correspond to the three subgroups [Formula: see text]. We consider the following situation: [Formula: see text] will be a rational surface and [Formula: see text] will be either a surface with [Formula: see text] or a K3 surface. The motivation for these assumptions is to have a strong control on the weight 2 Hodge-structure of the covering surface [Formula: see text], which allows us to study the infinitesimal Torelli property, the Chow groups and Chow motive, and the Tate and Mumford–Tate conjectures for [Formula: see text]. In particular, we classify all the bidouble covers [Formula: see text] satisfying the conditions above whenever [Formula: see text] is minimal; we obtain surfaces [Formula: see text] with [Formula: see text]. We also introduce another construction, called iterated bidouble cover, which allows us to obtain surfaces with higher value of [Formula: see text] for which we still have a strong control on the weight 2 Hodge-structure.
In this paper, we investigate a sequence of multi-bubble solutions {u(k)} to the following mean field equation: (-Delta)(n)u(k) = rho(k) h(x)e(uk)/integral(Omega)he(ukdx), in Omega, B(j)u(k) = 0, j=0, ..., n - 1, on partial derivative Omega, where Omega is a bounded and smooth domain in R-2n with n >= 2 as a positive integer, h is a C-2,C-beta positive function, rho(k) are constants such that 0 < rho(k) <= C, for some constant C and B-j, j = 0, ..., n - 1 stand for either Navier or Dirichlet boundary conditions. We show that (after passing to a subsequence if necessary) lim(k ->+infinity) rho(k) = 2(2n+1) n!pi(n)m for some positive integer m. Furthermore, we obtain the following sharp estimates of rho(k): rho(k) - 2(2n+1)n!pi(n)m = c(0) Sigma(m)(j=1)(h(p(k,j)))(-1/n) (epsilon k,j2)[1/2(2n+1)n!pi(n) Delta log h(p(k,j)) +Delta R-2n(p(k,j), p(k,j)) + Sigma(i not equal j)Delta G(2n)(p(k,j), p(k,i))] + o(Sigma(m)(j=1) is an element of(2)(k,j)), where c(0) is a positive generic constant, G(2n) is the Green function of (-Delta)(n) with either Navier or Dirichlet boundary conditions, R-2n is the regular part of G(2n), p(k,j) is the local maximum point of uk in a neighborhood of p(j) with p(j) as the blow-up point of {u(k)} for each j = 1, ..., m, and log (2n)!4(n) /is an element of(2n)(k,j) = u(k)(p(k,j)) - log(integral(Omega) he(uk)dx). Our approach extends the works of Chen-Lin [Sharp estimates for solutions of multi-bubbles in compact Riemann surfaces, Comm. Pure Appl. Math.55 (2002) 728-771] and Lin-Wei [Sharp estimates for bubbling solutions of a fourth order mean field equation, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5)6(4) (2007) 599-630], which studied the second-order and fourth-order equations, to the general even-order equation. Moreover, this result also hold for other boundary condition, if boundary blow-up can be excluded.
In this paper, we prove that flat free boundaries of solutions of the inhomogeneous one-phase Stefan problem are C-1 ,C-alpha. The method consists of employing a hodograph transform and deriving the regularity via a linearization technique, following the approach introduced by De Silva, Forcillo, and Savin in [D. De Silva, N. Forcillo and O. Savin, Perturbative estimates for the one-phase Stefan problem, Calc. Var. Partial Differential Equations 60(6) (2021) 219].
We investigate the boundary regularity of solutions to a class of variable-exponent gradient degenerate mixed fully nonlinear local-nonlo cal elliptic Dirichlet problems. A crucial feature of the operators under consideration is that they degenerate on the set of critical points, C := {x is an element of Omega : Du(x) = 0}. First, we establish the Lipschitz regularity of solutions using the Ishii-Lions viscosity method when the order of the fractional Laplacian, s is an element of (1/2, 1) (Theorem 1.5), under general conditions. Due to inapplicability of the comparison principle for the equations under consideration, the classical Perron's method for the existence of a solution cannot be employed. However, utilizing the Lipschitz estimates established in Theorem 1.5 and "vanishing viscosity" method, we prove the existence of a solution. Subsequently, we establish the interior C-1,C-delta-regularity of viscosity solutions using an improvement of the flatness technique when s is close enough to 1 (Theorem 1.6). Furthermore, under suitable assumptions, we establish the Ho & uml;lder regularity of solutions up to the boundary (Theorem 1.15), a result that is new even for analogous nonlo cal Dirichlet problems.
Using spectral flow, we provide a proof of [V. G. Kac, P. Moseneder Frajria and P. Papi, Unitarity of minimal W-algebras and their representations II: Ramond sector, Jpn. J. Math.20(2) (2025) 169-281, Theorem 9.17] on unitarity of Ramond twisted non-extremal representations of unitary minimal W-algebras that does not rely on the still conjectural exactness of the twisted quantum reduction functor (see Conjecture 9.11 of the same paper). When g=spo(2|2n), F(4), D(2,1;m/n), it is also proven that the unitarity of extremal (=massless) representations of the unitary minimal W-algebra W-min(k)(g) in the Ramond sector is equivalent to the unitarity of extremal representations in the Neveu-Schwarz sector.
We investigate the large time behavior of solutions to the initial-boundary value problem of the one-dimensional compressible Euler equations with space-dependent damping on the half line. It is shown that the unique solution of the compressible Euler equations with space-dependent damping globally exists and converges time-asymptotically to the diffusion wave at the optimal convergence rate, which coincides with the heat equation, and particularly improves the convergence rate of the recent work by Matsumura-Nishihara [Asymptotic behavior of solutions to the Cauchy problem for 1-D p-system with space dependent damping, SIAM J. Math. Anal. 56 (2024) 993-1015]. Compared to all the previous related works where either the damping coefficient is space-independent or the boundary effects are ignored, this gives the first optimal convergence rate for the compressible Euler equations with damping in the presence of both space-dependent damping and boundary effects. Our method of proof consists of the choice of new correction functions, the technical time-weighted energy estimates and the Green function method.
We establish Kato-Ponce inequalities for the mixed local and nonlo cal differential operator (-Delta) + (-Delta)(s/2 )and its inhomogeneous variant for a full range of Lebesgue indices including the endpoint case, and determine the sharp constraint condition of the regularity index. The homogeneous inequality applies over the same range as in the classical scenario, whereas the inhomogeneous inequality covers a broader range of indices. Furthermore, we derive sharp Kato-Ponce commutator estimates within this framework. Our proof is based on an adaption of Fourier analytic methods developed by Grafakos and Oh [L. Grafakos and S. Oh, The Kato-Ponce inequality, Commun. Partial Differ. Equ. 39 (2014) 1128-1157, doi:10.1080/03605302.2013.822885], Bourgain and Li [J. Bourgain and D. Li, On an endpoint Kato-Ponce inequality, Differ. Integr. Equ. 27 (2014) 1037-1072] and Oh and Wu [S. Oh and X. Wu, On L1 endpoint Kato-Ponce inequality, Math. Res. Lett. 27 (2020) 1129-1163, doi:10.4310/MRL.2020.v27.n4.a8] to the present setting, combined with a new lower bound estimate for the family of operators {delta(2-s)(delta(2)-Delta)(s/2)}(delta is an element of[0,1]).
Let M-i be a sequence of non-collapsed n-manifolds with two-sidedly bounded Ricci curvature. We show that the Gromov-Hausdorff limit space, Y, of the associated sequence of orthonormal frame bundles, FMi, equipped with an almost canonical metric, shares similar properties as a Ricci limit space of non-collapsing sequence i.e. the singular set has codimension >= 4 whose complement contains an open and dense C-1,C-alpha-Riemannian manifold.
In this paper, we study the Whittaker modules for the quantum enveloping algebra Uq(& sfr;& lfr;3) with respect to a fixed Whittaker function. We construct the universal Whittaker module, find all its Whittaker vectors and investigate the submodules generated by subsets of Whittaker vectors and corresponding quotient modules. We also determine the irreducibility of these quotient modules and show that they exhaust all irreducible Whittaker modules. Finally, we can determine all maximal submodules of the universal Whittaker module. The Whittaker model of Uq(& sfr;& lfr;3) is quite different from that of Uq(& sfr;& lfr;2) and finite-dimensional simple Lie algebras, since the center of our algebra is not a polynomial algebra.
In 2023, H. Brezis [Some of my favorite open problems, Rend. Lincei 34(2) (2023) 307-335] published a list of his "favorite open problems", which he described as challenges he had "raised throughout his career and has resisted so far". In this paper, we shall provide a partial answer to this question by presenting the existence of sign-changing solutions to the equation whenever the parameter is small enough. Our construction is based on the building blocks of Del Pino-Musso-Pacard-Pistoia sign-changing solutions to Yamabe problem.
We show that a uniformly Euclidean metric with isolated singularity on closed M-n=T-n#M-0, where 4 <= n <= 7 or n >= 4, M(0 )spin, and non-negative scalar curvature on the smooth part is flat and extends smoothly over the singularity. This confirms Schoen's Conjecture in these cases. The novel approach here, which is the key to the proof, is to show that the space has non-negative synthetic Ricci curvature, i.e. an RCD(0,n) space. Our result also holds when the singular set consists of a finite union of submanifolds (of possibly different dimensions) intersecting transversally under additional assumption on the co-dimension and the location of the singular set.
In this paper, let alpha is an element of & Ropf; and P-alpha(gamma) be the potential along the moment curve gamma = (t,& mldr;,t(n)) (t is an element of (0,infinity)) defined by setting, for any suitable function f, P(alpha)(gamma)f(x) :=integral(infinity)(0)f(x - gamma(t))t(alpha-1)dt, for all x is an element of & Ropf;(n). The authors obtain some equivalent characterizations via capacities of the embedding P alpha gamma from Lebesgue spaces L-p(& Ropf;(n)) associated with the n-dimensional Lebesgue measure to L-q(& Ropf;(n),mu) endowed with the non-negative Radon measure mu in the form of d mu(x) := w(x)dx for a weight w. The main ingredient of this paper lies in the different approach to the estimate of the capacity of the anisotropic cubes Q(R)(gamma )along gamma.
In this paper, we study the existence of bubbling solutions of the critical biharmonic problem with Navier boundary condition {Delta(2)u = |u|(p-1)u + & varepsilon;a(x)|u|(q-1)u in Omega, u = Delta u = 0 on partial derivative Omega, where a(x) is a smooth function, p = N+4 /N-4, q >= 1, & varepsilon; > 0, Omega is a smooth bounded domain in & Ropf;(N), N >= 5. More precisely, we obtain the existence of bubbling solutions concentrating at a single point in Omega. In the non-critical perturbation case q not equal p, the location of the concentrating point is closely related to the Robin function and the perturbed function a(x). However, the location of the concentrating point is determined by the critical point of a(x) in the critical perturbation case q = p.
In this paper, we consider two problems concerning the existence of travelling waves for a linear 1D wave equation with periodic acoustic impedance. The first one is the existence of a periodic travelling wave for a given acoustic impedance. The second one is the inverse problem of the existence of a periodic acoustic impedance function that supports a travelling wave with a prescribed amplitude. Some results about the regularity of the acoustic impedance function are given. The analytical findings are validated by numerical simulations.
We find the optimal function norm on the left-hand side of the mth order Sobolev-type inequality parallel to u parallel to Y ((Hn)) <= C parallel to del(m)(g)u parallel to X-(Hn) in the n-dimensional hyperbolic space H-n, 1 <= m < n. The optimal function norm in the inequality among all rearrangement-invariant function norms is completely characterized. A variety of concrete examples of optimal function norms is provided. The examples include delicate limiting cases and, especially when m >= 3, seem to provide new, improved inequalities in these limiting cases.
We study the global in time existence of small solutions to the Cauchy problem for the subcritical modified Korteweg-de Vries (mKdV) equation {partial derivative(t)u - 1/3 partial derivative(3)(x)u = t(nu)partial derivative(x)(u(3)), t > 0, x is an element of R, u(0,x) = u(0)(x), x is an element of R. We suppose that 0 < nu < 1/24.We remark that nu > 0 means that equation is subcritical in the sense of the large time asymptotic behavior of solutions. We assume that the initial data have an analytic extension on the sector and are small. Then we find the large time asymptotics of the solutions.
In this paper, we investigate the existence and decay properties of solutions to the following elliptic systems, which arise in the context of Bose-Einstein condensation: -Delta u(i )+ V-i(x)u(i )=& sum;(m)(j=1)g (ij)(x)|u(j)|(p)|u( i)|(p-2)u( i )+ mu(i)|u(i)|(2 & lowast;-2)u( i),i,j = 1, 2,& mldr;,m. To analyze the decay behavior of solutions, we use a variant of Moser's iteration technique, with general conditions imposed on the potentials V-i(x) and g(ij)(x). Our results extend recent findings by Angeles, Clapp and Salda & ntilde;a [Exponential decay of the solutions to nonlinear Schr & ouml;dinger systems, Calc. Var. Partial Differential Equations 62(5) (2023) 160]. We pay particular attention to potential classes that either vanish or are unbounded at infinity. The existence of ground states is established under these assumptions, and the derived decay estimates are also used to show that weak solutions exhibit either exponential or polynomial decay.
In this paper, we study two-dimensional Dirichlet problems (linear and nonlinear) with discontinuous coefficients, order one terms and data in L-1 (and no more).The focus is a Brezis-Merle inequality.
In this paper, we investigate the global well-posedness for cubic nonlinear Schr & ouml;dinger (NLS) equation i partial derivative(t)u + Delta(g)u = |u|(2)u posed on the three-dimensional compact manifolds (M,g) with initial data u(0) is an element of H-s(M) where s > root 21-1/ 4 for Zoll manifold and s > 1+3 root 5/ 8 for the product of spheres S-2 x S-1. We utilize the multilinear eigenfunction estimate on compact manifold to treat the interaction of different frequencies, which is more complicated compared to the case of flat torus [C. Fan, G. Staffilani, H. Wang and B. Wilson, On a bilinear Strichartz estimate on irrational tori, Anal. PDE 11 (2018) 919-944] and waveguide manifold [Z. Zhao and J. Zheng, Long time dynamics for defocusing cubic nonlinear Schr & ouml;dinger equations on three dimensional product space, SIAM J. Math. Anal. 53 (2020) 3644-3660]. Moreover, combining with the I-method adapted to the non-periodic case, bilinear Strichartz estimates along with the scale-invariant Lp linear Strichartz estimates, we partially obtain the similar result of [Z. Zhao and J. Zheng, Long time dynamics for defocusing cubic nonlinear Schr & ouml;dinger equations on three dimensional product space, SIAM J. Math. Anal. 53 (2020) 3644-3660] on non-flat compact manifold setting. As a consequence, we obtain the polynomial bounds of the H-s norm of solution u.
