On measure‐valued solutions to a two‐dimensional gravity‐driven avalanche flow model

This paper concerns measure‐valued solutions for the two‐dimensional granular avalanche flow model introduced by Savage and Hutter. The system is similar to the isentropic compressible Euler equations, except for a Coulomb–Mohr friction law in the source term. We will partially follow the study of measure‐valued solutions given by DiPerna and Majda. However, due to the multi‐valued nature of the friction law, new more sensitive measures must be introduced. The main idea is to consider the class of x‐dependent maximal monotone graphs of non‐single‐valued operators and their relation with 1‐Lipschitz, Carathéodory functions. This relation allows to introduce generalized Young measures for x‐dependent maximal monotone graph. Copyright © 2005 John Wiley & Sons, Ltd.     

A Classification of Isolated Singularities of Elliptic Monge‐Ampére Equations in Dimension Two

Let ?₁ denote the space of solutions z(x,y) to an elliptic, real analytic Monge‐Ampére equation whose graphs have a non‐removable isolated singularity at the origin. We prove that ?₁ is in one‐to‐one correspondence with ?₂ × ?₂, where ?₂ is a suitable subset of the class of regular, real analytic, strictly convex Jordan curves in ?₂. We also describe the asymptotic behavior of solutions of the Monge‐Ampére equation in the C^k‐smooth case, and a general existence theorem for isolated singularities of analytic solutions of the more general equation .© 2015 Wiley Periodicals, Inc.     

The fast signal diffusion limit in a chemotaxis system with strong signal sensitivity

This paper gives an insight into making a mathematical bridge between the parabolic‐parabolic signal‐dependent chemotaxis system and its parabolic‐elliptic version. To be more precise, this paper deals with convergence of a solution for the parabolic‐parabolic chemotaxis system with strong signal sensitivity to that for the parabolic‐elliptic chemotaxis system where Ω is a bounded domain in () with smooth boundary, is a constant and χ is a function generalizing In chemotaxis systems parabolic‐elliptic systems often gave some guide to methods and results for parabolic‐parabolic systems. However, the relation between parabolic‐elliptic systems and parabolic‐parabolic systems has not been studied except for the case that . Namely, in the case that Ω is a bounded domain, it still remains to analyze on the following question: Does a solution of the parabolic‐parabolic system converge to that of the parabolic‐elliptic system as ? This paper gives some positive answer in the chemotaxis system with strong signal sensitivity.     

Independence,odd girth,and average degree

We prove several tight lower bounds in terms of the order and the average degree for the independence number of graphs that are connected and/or satisfy some odd girth condition. Our main result is the extension of a lower bound for the independence number of triangle‐free graphs of maximum degree at most three due to Heckman and Thomas [Discrete Math 233 (2001), 233–237] to arbitrary triangle‐free graphs. For connected triangle‐free graphs of order n and size m, our result implies the existence of an independent set of order at least (4n?m?1)/7. © 2010 Wiley Periodicals, Inc. J Graph Theory 67:96‐111, 2011     

On the absence of eigenvalues of Maxwell and Lamé systems in exterior domains

The arguments showing non‐existence of eigensolutions to exterior‐boundary value problems associated with systems—such as the Maxwell and Lamé system—rely on showing that such solutions would have to have compact support and therefore—by a unique continuation property—cannot be non‐trivial. Here we will focus on the first part of the argument. For a class of second order elliptic systems it will be shown that L²‐solutions in exterior domains must have compact support. Both the asymptotically isotropic Maxwell system and the Lamé system with asymptotically decaying perturbations can be reduced to this class of elliptic systems. Copyright © 2004 John Wiley & Sons, Ltd.     

Gradient estimates and Harnack inequalities for nonlinear parabolic and nonlinear elliptic equations on Riemannian manifolds

We derive the gradient estimates and Harnack inequalities for positive solutions of nonlinear parabolic and nonlinear elliptic equations (Δ − ∂/∂t) u(x, t) + h(x, t)u^α(x, t) = 0 and Δu + b · u + hu^α = 0 on Riemannian manifolds. We also obtain a theorem of Liouville type for positive solutions of the nonlinear elliptic equation.     

Graphs Between the Elliptic and Parabolic Harnack Inequalities

We present graphs that satisfy the uniform elliptic Harnack inequality, for harmonic functions, but not the stronger parabolic one, for solutions of the discrete heat equation. It is known that the parabolic Harnack inequality is equivalent to the conjunction of a volume regularity and a L ² Poincaré inequality. The first example of graph satisfying the elliptic but not the parabolic Harnack inequality is due to M. Barlow and R. Bass. It satisfies the volume regularity and not the Poincaré inequality. We construct another example that does not satisfy the volume regularity.     

Gradient estimates for parabolic systems from composite material

In this paper, we derive W~(1,∞) and piecewise C~(1,α) estimates for solutions, and their t-derivatives, of divergence form parabolic systems with coefficients piecewise H¨older continuous in space variables x and smooth in t. This is an extension to parabolic systems of results of Li and Nirenberg [Comm Pure Appl Math, 2003, 56:892–925] on elliptic systems. These estimates depend on the shape and the size of the surfaces of discontinuity of the coefficients, but are independent of the distance between these surfaces.     

Singular solutions of elliptic equations with nonstandard growth

We prove a removability result for nonlinear elliptic equations withp (x)‐type nonstandard growth and estimate the growth of solutions near a nonremovable isolated singularity. To accomplish this, we employ a Harnack estimate for possibly unbounded solutions and the fact that solutions with nonremovable isolated singularities are p (x)‐superharmonic functions (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Cyclic bi‐embeddings of Steiner triple systems on 12s + 7 points

A cyclic face 2‐colourable triangulation of the complete graph K_n in an orientable surface exists for n ≡ 7 (mod 12). Such a triangulation corresponds to a cyclic bi‐embedding of a pair of Steiner triple systems of order n, the triples being defined by the faces in each of the two colour classes. We investigate in the general case the production of such bi‐embeddings from solutions to Heffter's first difference problem and appropriately labelled current graphs. For n = 19 and n = 31 we give a complete explanation for those pairs of Steiner triple systems which do not admit a cyclic bi‐embedding and we show how all non‐isomorphic solutions may be identified. For n = 43 we describe the structures of all possible current graphs and give a more detailed analysis in the case of the Heawood graph. © 2002 Wiley Periodicals, Inc. J Combin Designs 10: 92–110, 2002; DOI 10.1002/jcd.10001     

Ritz–Galerkin method for solving a parabolic equation with non‐local and time‐dependent boundary conditions

The paper is devoted to the investigation of a parabolic partial differential equation with non‐local and time‐dependent boundary conditions arising from ductal carcinoma in situ model. Approximation solution of the present problem is implemented by the Ritz–Galerkin method, which is a first attempt at tackling parabolic equation with such non‐classical boundary conditions. In the process of dealing with the difficulty caused by integral term in non‐local boundary condition, we use a trick of introducing the transition function G(x,t) to convert non‐local boundary to another non‐classical boundary, which can be handled with the Ritz–Galerkin method. Illustrative examples are included to demonstrate the validity and applicability of the technique in this paper. Copyright © 2015 John Wiley & Sons, Ltd.     

Blow‐up phenomena in chemotaxis systems with a source term

This paper deals with a parabolic–parabolic Keller–Segel‐type system in a bounded domain of , {N = 2;3}, under different boundary conditions, with time‐dependent coefficients and a positive source term. The solutions may blow up in finite time t^?; and under appropriate assumptions on data, explicit lower bounds for blow‐up time are obtained when blow up occurs. Copyright © 2015 John Wiley & Sons, Ltd.     

Asymptotic analysis of solutions to parabolic systems

We study asymptotics as t → ∞ of solutions to a linear, parabolic system of equations with time‐dependent coefficients in Ω × (0, ∞), where Ω is a bounded domain. On ? Ω × (0, ∞) we prescribe the homogeneous Dirichlet boundary condition. For large values of t, the coefficients in the elliptic part are close to time‐independent coefficients in an integral sense which is described by a certain function κ (t). This includes in particular situations when the coefficients may take different values on different parts of Ω and the boundaries between them can move with t but stabilize as t → ∞. The main result is an asymptotic representation of solutions for large t. As a corollary, it is proved that if κ ∈ L¹(0, ∞), then the solution behaves asymptotically as the solution to a parabolic system with time‐independent coefficients (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Uniqueness and nonuniqueness of the Stokes and Oseen flows

The uniqueness class for the solutions to the Cauchy problem for flows modeled by the time-dependent Stokes and Oseen systems of equations is determined as in the growth class C exp(α|x|²). An example of a type considered by Tychonoff [Mat. Sb. 42 (1935), 199–216] is given that establishes the lack of uniqueness for such Stokes and Oseen systems. Even for the incompressible Navier–Stokes system, an example shows that rapidly growing nonphysical mathematical solutions exist.     

Global Regularity of Weak Solutions to Quasilinear Elliptic and Parabolic Equations with Controlled Growth

We establish global regularity for weak solutions to quasilinear divergence form elliptic and parabolic equations over Lipschitz domains with controlled growth conditions on low order terms. The leading coefficients belong to the class of BMO functions with small mean oscillations with respect to x.     

Exact elliptic compactons in generalized Korteweg–De Vries equations

Using the action principle, and assuming a solitary wave of the generic form u(x,t) = AZ(β(x + q(t)), we derive a general theorem relating the energy, momentum, and velocity of any solitary wave solution of the generalized Korteweg‐De Vries equation K*(l,p). Specifically we find that , where l,p are nonlinearity parameters. We also relate the amplitude, width, and momentum to the velocity of these solutions. We obtain the general condition for linear and Lyapunov stability. We then obtain a two‐parameter family of exact solutions to these equations, which include elliptic and hyper‐elliptic compacton solutions. For this general family we explicitly verify both the theorem and the stability criteria. © 2006 Wiley Periodicals, Inc. Complexity 11: 30–34, 2006     

An extension of Jörgens–Calabi–Pogorelov theorem to parabolic Monge–Ampère equation

We extend a theorem of Jörgens, Calabi and Pogorelov on entire solutions of elliptic Monge–Ampère equation to parabolic Monge–Ampère equation, and obtain delicate asymptotic behavior of solutions at infinity. For the dimension \(n\ge 3\), the work of Gutiérrez and Huang in Indiana Univ. Math. J. 47, 1459–1480 (1998) is an easy consequence of our result. And along the line of approach in this paper, we can treat other parabolic Monge–Ampère equations.     

A characterization of weakly four‐connected graphs

A graph G = (V, E) is called weakly four‐connected if G is 4‐edge‐connected and G ? x is 2‐edge‐connected for all x ∈ V. We give sufficient conditions for the existence of ‘splittable’ vertices of degree four in weakly four‐connected graphs. By using these results we prove that every minimally weakly four‐connected graph on at least four vertices contains at least three ‘splittable’ vertices of degree four, which gives rise to an inductive construction of weakly four‐connected graphs. Our results can also be applied in the problem of finding 2‐connected orientations of graphs. © 2006 Wiley Periodicals, Inc. J Graph Theory 52: 217–229, 2006     

On constructible graphs,locally Helly graphs,and convexity

A (finite or infinite) graph G is constructible if there exists a well‐ordering ≤ of its vertices such that for every vertex x which is not the smallest element, there is a vertex y < x which is adjacent to x and to every neighbor z of x with z < x. Particular constructible graphs are Helly graphs and connected bridged graphs. In this paper we study a new class of constructible graphs, the class of locally Helly graphs. A graph G is locally Helly if, for every pair (x,y) of vertices of G whose distance is d ≥ 2, there exists a vertex whose distance to x is d ? 1 and which is adjacent to y and to all neighbors of y whose distance to x is at most d. Helly graphs are locally Helly, and the converse holds for finite graphs. Among different properties we prove that a locally Helly graph is strongly dismantable, hence cop‐win, if and only if it contains no isometric rays. We show that a locally Helly graph G is finitely Helly, that is, every finite family of pairwise non‐disjoint balls of G has a non‐empty intersection. We give a sufficient condition by forbidden subgraphs so that the three concepts of Helly graphs, of locally Helly graphs and of finitely Helly graphs are equivalent. Finally, generalizing different results, in particular those of Bandelt and Chepoi 1 about Helly graphs and bridged graphs, we prove that the Helly number h(G) of the geodesic convexity in a constructible graph G is equal to its clique number ω(G), provided that ω(G) is finite. © 2003 Wiley Periodicals, Inc. J Graph Theory 43: 280–298, 2003     

On viscosity solutions for nontotally parabolic fully nonlinear equations

It is shown how to prove global unique solvability of the first initial-boundary value problem in the class of continuous viscosity solutions for some classes of equations −u_t+F(u_x,u_xx)=g(x, t, u_x), where F(p, A) is elliptic only on some nonlinear subsets of values of the arguments (p, A). For this purpose we use the techniques developed in the theory of viscosity solutions for degenerate elliptic equations. Bibliography: 12 titles. Published inZapiski Nauchnykh Seminarov POMI, Vol. 233, 1996, pp. 112–130.