Supremum principles for the higher-order derivatives of solutions to the Dirichlet problem for parabolic equations without compatibility conditions between the data

The Dirichlet problem is considered for the heat equation ut=auxx, a>0 a constant, for (x,t)∈[0,1]×[0,T], without assuming any compatibility condition between initial and boundary data at the corner points (0,0) and (1,0). Under some smoothness restrictions on the data (stricter than those required by the classical maximum principle), weak and strong supremum and infimum principles are established for the higher-order derivatives, ut and uxx, of the bounded classical solutions. When compatibility conditions of zero order are satisfied (i.e., initial and boundary data coincide at the corner points), these principles allow to estimate the higher-order derivatives of classical solutions uniformly from below and above on the entire domain, except that at the two corner points. When compatibility conditions of the second order are satisfied (i.e., classical solutions belong to on the closed domain), the results of the paper are a direct consequence of the classical maximum and minimum principles applied to the higher-order derivatives. The classical principles for the solutions to the Dirichlet problem with compatibility conditions are generalized to the case of the same problem without any compatibility condition. The Dirichlet problem without compatibility conditions is then considered for general linear one-dimensional parabolic equations. The previous results as well as some new properties of the corresponding Green functions derived here allow to establish uniformL¹-estimates for the higher-order derivatives of the bounded classical solutions to the general problem.     

On the stability of the Serrin problem

We investigate stability issues concerning the radial symmetry of solutions to Serrin's overdetermined problems. In particular, we show that, if u is a solution to Δu=n in a smooth domain Ω⊂Rn, u=0 on ∂Ω and |Du| is “close” to 1 on ∂Ω, then Ω is “close” to the union of a certain number of disjoint unitary balls.     

A Pizzetti-type formula for the heat operator

We provide an asymptotic expansion of the integral mean of a smooth function over the Heat ball. Namely we generalize to the Heat operator the so-called Pizzetti’s Formula, which expresses the integral mean of a smooth function over an Euclidean ball in terms of a power series with respect to the radius of the ball having the iterated of the ordinary Laplace operator as coefficients. Similarly here, we express the heat integral mean as a power series with respect to the radius of the heat ball, whose coefficients are powers of a distorted heat operator. We also discuss sufficient conditions to have a finite sum. Received: 27 May 2005     

Global solvability for first order real linear partial differential operators

F. Treves, in [17], using a notion of convexity of sets with respect to operators due to B. Malgrange and a theorem of C. Harvey, characterized globally solvable linear partial differential operators on C^∞(X), for an open subset X of Rn.Let P=L+c be a linear partial differential operator with real coefficients on a C^∞ manifold X, where L is a vector field and c is a function. If L has no critical points, J. Duistermaat and L. Hörmander, in [2], proved five equivalent conditions for global solvability of P on C^∞(X).Based on Harvey-Treves's result we prove sufficient conditions for the global solvability of P on C^∞(X), in the spirit of geometrical Duistermaat-Hörmander's characterizations, when L is zero at precisely one point. For this case, additional non-resonance type conditions on the value of c at the equilibrium point are necessary.     

How big can the circuits of a bridge of a maximal circuit be?

IfCE(G) is a maximum cardinality cocircuit of a 2-connected graphG, then no other maximum cocircuit is contained in one and the same block ofG-C. The analogous conjecture for real representable matroids would have important applications to classifying convex bodies with a certain Helly type property.     

H 1-perturbations of Smooth Solutions for a Weakly Dissipative Hyperelastic-rod Wave Equation

We consider a weakly dissipative hyperelastic-rod wave equation (or weakly dissipative Camassa–Holm equation) describing nonlinear dispersive dissipative waves in compressible hyperelastic rods. We fix a smooth solution and establish the existence of a strongly continuous semigroup of global weak solutions for any initial perturbation from In particular, the supersonic solitary shock waves [8] are included in the analysis. Dedicated to the memory of Professor Aldo Cossu The research of K.H. Karlsen is supported by an Outstanding Young Investigators Award from the Research Council of Norway. The current address of G.M. Coclite is Department of Mathematics, University of Bari, Via E. Orabona 4, 70125 Bari, Italy     

On the ground state eigenfunction of a convex domain in Euclidean space

We study the first eigenfunction ₁ of the Dirichlet Laplacian on a convex domain in Euclidean space. Elementary properties of Bessel functions yield that if D is a sector in Euclidean plane with area 1 and the angle tends to 0. We aim to characterize those domains D such that is large in terms of the ratio of the first eigenvalue of D and the infimum of the first eigenvalues of all subdomains D of D with given volume.Research supported by the Deutsche Forschungsgemeinschaft     

Well-posedness of higher-order Camassa-Holm equations

We consider higher-order Camassa-Holm equations describing exponential curves of the manifold of smooth orientation-preserving diffeomorphisms of the unit circle in the plane. We establish the existence of global weak solutions. We also present some invariant spaces under the action of the equation. Moreover, we prove a “weak equals strong” uniqueness result.     

Cauchy problem for p-evolution equations

We consider p-evolution equations with real characteristics. We give a condition, on the lower order terms, that is sufficient for well-posedness of the Cauchy problem in Sobolev spaces.     

Splitting of resonant and scattering frequencies under shape deformation

It is well known that the main difficulty in solving eigenvalue problems under shape deformation relates to the continuation of multiple eigenvalues of the unperturbed configuration. These eigenvalues may evolve, under shape deformation, as separated, distinct eigenvalues, and the splitting may only become apparent at high orders in their Taylor expansion. In this paper, we address the splitting problem in the evaluation of resonant and scattering frequencies of the two-dimensional Laplacian operator under boundary variations of the domain. By using surface potentials we show that the eigenvalues are the characteristic values of meromorphic operator-valued functions that are of Fredholm type with index 0. We then proceed from the generalized Rouché's theorem to investigate the splitting problem.     

On A Hypergraph Turán Problem Of Frankl

Let be the 2k-uniform hypergraph obtained by letting P₁, . . .,P_r be pairwise disjoint sets of size k and taking as edges all sets P_i∪P_j with i ≠ j. This can be thought of as the ‘k-expansion’ of the complete graph K_r: each vertex has been replaced with a set of size k. An example of a hypergraph with vertex set V that does not contain can be obtained by partitioning V = V1 ∪V₂ and taking as edges all sets of size 2k that intersect each of V₁ and V₂ in an odd number of elements. Let denote a hypergraph on n vertices obtained by this construction that has as many edges as possible. For n sufficiently large we prove a conjecture of Frankl, which states that any hypergraph on n vertices that contains no has at most as many edges as . Sidorenko has given an upper bound of for the Tur′an density of for any r, and a construction establishing a matching lower bound when r is of the form 2^p+1. In this paper we also show that when r=2^p+1, any -free hypergraph of density looks approximately like Sidorenko’s construction. On the other hand, when r is not of this form, we show that corresponding constructions do not exist and improve the upper bound on the Turán density of to , where c(r) is a constant depending only on r. The backbone of our arguments is a strategy of first proving approximate structure theorems, and then showing that any imperfections in the structure must lead to a suboptimal configuration. The tools for its realisation draw on extremal graph theory, linear algebra, the Kruskal–Katona theorem and properties of Krawtchouck polynomials. * Research supported in part by NSF grants DMS-0355497, DMS-0106589, and by an Alfred P. Sloan fellowship.     

Global existence and convergence rates of smooth solutions for the compressible magnetohydrodynamic equations

In this paper, we are concerned with the global existence and convergence rates of the smooth solutions for the compressible magnetohydrodynamic equations in R³. We prove the global existence of the smooth solutions by the standard energy method under the condition that the initial data are close to the constant equilibrium state in H³-framework. Moreover, if additionally the initial data belong to L^p with , the optimal convergence rates of the solutions in L^q-norm with 2≤q≤6 and its spatial derivatives in L²-norm are obtained.     

Spatial critical points not moving along the heat flow II: The centrosymmetric case

We consider solutions of initial-boundary value problems for the heat equation on bounded domains in and their spatial critical points as in the previous paper [MS]. In Dirichlet, Neumann, and Robin homogeneous initial-boundary value problems on bounded domains, it is proved that if the origin is a spatial critical point never moving for sufficiently many compactly supported initial data being centrosymmetric with respect to the origin, then the domain must be centrosymmetric with respect to the origin. Furthermore, we consider spatial zero points instead of spatial critical points, and prove some similar symmetry theorems. Also, it is proved that these symmetry theorems hold for initial-boundary value problems for the wave equation. Received October 31, 1997; in final form February 3, 1998     

Steiner Numbers in Graphs

Abstract

The Steiner distance d(S) of a set S of vertices in a connected graph G is the minimum size of a connected subgraph of G that contains S. The Steiner number s(G) of a connected graph G of order p is the smallest positive integer m for which there exists a set S of m vertices of G such that d(S) = p—1. A smallest set S of vertices of a connected graph G of order p for which d(S) = p—1 is called a Steiner spanning set of G. It is shown that every connected graph has a unique Steiner spanning set. If G is a connected graph of order p and k is an integer with 0 ≤ k ≤ p—1, then the kth Steiner number sk(G) of G is the smallest positive integer m for which there exists a set S of m vertices of G such that d(S) = k. The sequence s_o(G),s₁ (G),…,8_p-1(G) is called the Steiner sequence of G. Steiner sequences for trees are characterized.     

Deduction of L. Hörmander's extension of Ásgeirsson's mean value theorem

L. Hörmander's extension of Ásgeirsson's mean value theorem states that if u is a solution of the inhomogeneous ultrahyperbolic equation (Δ_x−Δ_y)u=f, , , then

     

Expanding graphs contain all small trees

The assertion of the title is formulated and proved. The result is then used to construct graphs with a linear number of edges that, even after the deletion of almost all of their edges or almost all of their vertices, continue to contain all small trees.     

Comparison principle for non-cooperative elliptic systems

This paper presents some sufficient conditions for the validity of the comparison principle for weak and classical solutions of non-cooperative weakly-coupled systems of elliptic second-order PDE.     

Stability of a Generalized Sobolev System

The linear stability problem of the rotational motion of a rigid body around a fixed point containing an inner cavity filled up with an ideal fluid is considered. In this paper, we also assume that the fluid is rotating. The effect of the angular velocities of the rigid body and the fluid in the stability problem is studied. The case of a cavity ellipsoidal is presented in detail.