Alexander Grigor’yanDepartment of Mathematics, University of Bielefeld, 33501, Bielefeld, GermanyYong LinDepartment of Mathematics, Renmin University of China, Beijing, 100872, People’s Republic of ChinaYunyan YangDepartment of Mathematics, Renmin University of China, Beijing, 100872, People’s Republic of China
Analysis of PDEsmathscidoc:2207.03001
Calculus of Variations and Partial Differential Equations, 55, (92), 2016.7
Let G=(V,E) be a connected finite graph and Δ be the usual graph Laplacian. Using the calculus of variations and a method of upper and lower solutions, we give various conditions such that the Kazdan–Warner equation Δu=c−he^u has a solution on V, where c is a constant, and h:V→R is a function. We also consider similar equations involving higher order derivatives on graph. Our results can be compared with the original manifold case of Kazdan and Warner (Ann. Math. 99(1):14–47, 1974).
We study the homogenization of a stationary conductivity problem in a random heterogeneous medium with highly oscillating conductivity coefficients and an ensemble of simply closed conductivity resistant membranes. This medium is randomly deformed and then rescaled from a periodic one with periodic membranes, in a manner similar to the random medium proposed by Blanc, Le Bris, and Lions (2006). Across the membranes, the flux is continuous but the potential field itself undergoes a jump of Robin-type. We prove that, for almost all realizations of the random deformation, as the small scale of variations of the medium goes to zero, the random conductivity problem is well approximated by that of an effective medium which has deterministic and constant coefficients and contains no membrane. The effective coefficients are explicitly represented. One of our main contributions is to provide a solution to the associated auxiliary problem that is posed on the whole space with infinitely many interfaces, a setting that is out of the standard stationary ergodic framework.
Wenjia JingYau Mathematical Sciences Center, Tsinghua University, No.1 Tsinghua Yuan, Beijing 100084, ChinaOlivier PinaudDepartment of Mathematics, Colorado State University, Fort Collins, CO 80525, USA
Analysis of PDEsmathscidoc:2206.03013
Discrete and Continuous Dynamical Systems - B, 24, (10), 5377-5407, 2019.10
This work concerns the analysis of wave propagation in random media. Our medium of interest is sea ice, which is a composite of a pure ice background and randomly located inclusions of brine and air. From a pulse emitted by a source above the sea ice layer, the main objective of this work is to derive a model for the backscattered signal measured at the source/detector location. The problem is difficult in that, in the practical configuration we consider, the wave impinges on the layer with a non-normal incidence. Since the sea ice is seen by the pulse as an effective (homogenized) medium, the energy is specularly reflected and the backscattered signal vanishes in a first order approximation. What is measured at the detector consists therefore of corrections to leading order terms, and we focus in this work on the homogenization corrector. We describe the propagation by a random Helmholtz equation, and derive an expression of the corrector in this layered framework. We moreover obtain a transport model for quadratic quantities in the random wavefield in a high frequency limit.
Wenjia JingYau Mathematical Sciences Center, Tsinghua University, No.1 Tsinghua Yuan, Beijing 100084, ChinaHiroyoshi MitakeGraduate School of Mathematical Sciences, University of Tokyo 3-8-1 Komaba, Meguro-ku, Tokyo, 153-8914, JapanHung V. TranDepartment of Mathematics, University of Wisconsin Madison, Van Vleck hall, 480 Lincoln drive, Madison, WI 53706, USA
Analysis of PDEsmathscidoc:2206.03015
Journal of Differential Equations, 268, (6), 2886-2909, 2020.3
We study a generalized ergodic problem (E), which is a Hamilton-Jacobi equation of contact type, in the flat n-dimensional torus. We first obtain existence of solutions to this problem under quite general assumptions. Various examples are presented and analyzed to show that (E) does not have unique solutions in general. We then study uniqueness structures of solutions to (E) in the convex setting by using the nonlinear adjoint method.
Let G=(V,E) be a locally finite connected weighted graph, and Δ be the usual graph Laplacian. In this article, we study blow-up problems for the nonlinear parabolic equation u_t = Δu + f(u) on G. The blow-up phenomenons for u_t = Δu + f(u) are discussed in terms of two cases: (i) an initial condition is given; (ii) a Dirichlet boundary condition is given. We prove that if f satisfies appropriate conditions, then the corresponding solutions will blow up in a finite time.