Search

In this paper, we use Aleksandrov's reflection principle to prove symmetry of solutions to F (∇ S n 2 u + u I) = f (u , u 2 + | ∇ S n u | 2 ) , where u is the support function of a convex body, and F is a function of principal radii. As a corollary, alongside [Ivaki, arXiv:2307.06252 ], we provide an alternative proof of the uniqueness of solution to the isotropic Gaussian-Minkowski problem. [ABSTRACT FROM AUTHOR]

In this paper, we establish an existence and uniqueness result for the system of quasilinear stochastic partial differential equations (SPDEs for short) with reflection in a convex domain in R K by analytical approach. The method is based on the approximation of the penalized systems of SPDEs. [ABSTRACT FROM AUTHOR]

Consider the semilinear elliptic equation − Δ u = λ f (u) in the unit ball B 1 ⊂ R N , with Dirichlet data u | ∂ B 1 = 0 , where λ ≥ 0 is a real parameter and f is a C 1 positive, nondecreasing and convex function in [ 0 , ∞) such that f (s) / s → ∞ as s → ∞. In this paper we study the behavior of f ′ (u ⁎) near the origin when u ⁎ , the extremal solution of the previous problem associated to λ = λ ⁎ , is singular. This answers to an open problems posed by Brezis and Vázquez [2, Open problem 5]. [ABSTRACT FROM AUTHOR]

In this paper we propose a primal-dual dynamical approach to the minimization of a structured convex function consisting of a smooth term, a nonsmooth term, and the composition of another nonsmooth term with a linear continuous operator. In this scope we introduce a dynamical system for which we prove that its trajectories asymptotically converge to a saddle point of the Lagrangian of the underlying convex minimization problem as time tends to infinity. In addition, we provide rates for both the violation of the feasibility condition by the ergodic trajectories and the convergence of the objective function along these ergodic trajectories to its minimal value. Explicit time discretization of the dynamical system results in a numerical algorithm which is a combination of the linearized proximal method of multipliers and the proximal ADMM algorithm. [ABSTRACT FROM AUTHOR]

Abstract In this paper we rigorously justify the propagation of chaos for the parabolic–elliptic Keller–Segel equation over bounded convex domains. The boundary condition under consideration is the no-flux condition. As intermediate steps, we establish the well-posedness of the associated stochastic equation as well as the well-posedness of the Keller–Segel equation for bounded weak solutions. [ABSTRACT FROM AUTHOR]

This paper studies traveling fronts to cooperation–diffusion systems in R N for N ≥ 3 . We consider ( N − 2 ) -dimensional smooth surfaces as boundaries of strictly convex compact sets in R N − 1 , and define an equivalence relation between them. We prove that there exists a traveling front associated with a given surface and show its stability. The associated traveling fronts coincide up to phase transition if and only if the given surfaces satisfy the equivalence relation. [ABSTRACT FROM AUTHOR]

Abstract: Let be a non-constant T-periodic solution to the ordinary differential equation in a Banach space X where f is assumed to be Lipschitz continuous with constant L. Then there exists a constant c such that , with c only depending on X. It is known that in any Banach space and that in any Hilbert space, but whereas the bound of is sharp in any Hilbert space, there exists only one known example of a Banach space such that is optimal. In this paper, we show that the inequality is in fact strict in any strictly convex Banach space. Moreover, we improve the lower bound for and for a range of p close to by using a form of Wirtinger's inequality for functions in . [Copyright &y& Elsevier]

Abstract: In this paper we deduce a formula for the fractional Laplace operator on radially symmetric functions useful for some applications. We give a criterion of subharmonicity associated with , and apply it to a problem related to the Hessian inequality of Sobolev type: where is the k-Hessian operator on , , under some restrictions on a k-convex function u. In particular, we show that the class of u for which the above inequality was established in Ferrari et al. contains the extremal functions for the Hessian Sobolev inequality of X.-J. Wang (1994) . This is proved using logarithmic convexity of the Gaussian ratio of hypergeometric functions which might be of independent interest. [Copyright &y& Elsevier]

Abstract: The approach based on the construction of some nonlinear functionals was proved to be robust in the study of the well-posedness theories of hyperbolic conservation laws, especially in one space dimensional case. In particular, a generalized entropy functional was constructed in [T.-P. Liu, T. Yang, A new entropy functional for scalar conservation laws, Comm. Pure Appl. Math. 52 (1999) 1427–1442] for the stability of weak solutions. However, this generalized functional is so far only defined for scalar equations with convex flux function. In this paper, we introduce a new nonlinear functional which is motivated by the new Glimm functional introduced in [J.-L. Hua, Z.-H. Jiang, T. Yang, A new Glimm functional and convergence rate of Glimm scheme for general systems of hyperbolic conservation laws, preprint] for general scalar conservation laws. This functional improves the one given in [H.-X. Liu, T. Yang, A nonlinear functional for general scalar hyperbolic conservation laws, J. Differential Equations 235 (2) (2007) 658–667] and it can be viewed as a better attempt for the generalized entropy functional for general equations. [Copyright &y& Elsevier]