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 G := (& Zopf;/2 & Zopf;)(2)-Galois cover X -> Y. In this situation, there exist three intermediate quotients Y1,Y2 and Y-3 which correspond to the three subgroups & Zopf;/2 & Zopf; <= G. We consider the following situation: Y will be a rational surface and Yi will be either a surface with pg = 0 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 X, which allows us to study the infinitesimal Torelli property, the Chow groups and Chow motive, and the Tate and Mumford-Tate conjectures for X. In particular, we classify all the bidouble covers X -> Y satisfying the conditions above whenever Y is minimal; we obtain surfaces X with p(g)(X) = 1, 2, 3. We also introduce another construction, called iterated bidouble cover, which allows us to obtain surfaces with higher value of p(g) 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.
