The initial boundary value problem for the nonlinear parabolic equation in unbounded domain

In this paper we consider the mixed problem for the equation u _tt  + A ₁ u + A ₂(u _t ) + g(u _t ) = f(x, t) in unbounded domain, where A ₁ is a linear elliptic operator of the fourth order and A ₂ is a nonlinear elliptic operator of the second order. Under natural assumptions on the equation coefficients and f we proof existence of a solution. This result contains, as a special case, some of known before theorems of existence. Essentially, in difference up to previous results we prove theorems of existence without the additional assumption on behavior of solution at infinity.      

Some theorems in affine differential geometry

In this paper we prove that an affine hypersphere with scalar curvature zero in a unimodular affine space of dimensionn+1 must be contained either in an elliptic paraboloid or in an affine image of the hypersurfacex ₁ x ₂...x _n+1=const. We prove also that an affine complete, affine maximal surface is an elliptic paraboloid if its affine normals omit 4 or more directions in general position.The Project Supported by National Natural Science Foundation of China     

Kac–Moody $${\widetilde{E}_k}$$-bundles over elliptic curves and del Pezzo surfaces with singularities of type A

Given an elliptic curve Σ, flat E _k -bundles over Σ are in one-to-one correspondence with smooth del Pezzo surfaces of degree 9 − k containing Σ as an anti-canonical curve. This correspondence was generalized to Lie groups of any type. In this article, we show that there is a similar correspondence between del Pezzo surfaces of degree 0 with an A _d -singularity containing Σ as an anti-canonical curve and Kac–Moody [(E)\tilde]_k{\widetilde{E}_{k}}-bundles over Σ with k = 8 − d. In the degenerate case where surfaces are rational elliptic surfaces, the corresponding [(E)\tilde]_k{\widetilde{E}_k}-bundles over Σ can be reduced to E _k -bundles.     

On regularity of a boundary point for higher-order parabolic equations: towards Petrovskii-type criterion by blow-up approach

The classic problem of regularity of boundary points for higher-order partial differential equations (PDEs) is concerned. For second-order elliptic and parabolic equations, this study was completed by Wiener’s (J. Math. Phys. Mass. Inst. Tech. 3:127–146, 1924) and Petrovskii’s (Math. Ann. 109:424–444, 1934) criteria, and was extended to more general equations including quasilinear ones. Since the 1960–1970s, the main success was achieved for 2mth-order elliptic PDEs; e.g., by Kondrat’ev and Maz’ya. However, the higher-order parabolic ones, with infinitely oscillatory kernels, were not studied in such details. As a basic model, explaining typical difficulties of regularity issues, the 1D bi-harmonic equation in a domain shrinking to the origin (0, 0) is concentrated upon:

Saddle solutions of nonlinear elliptic equations involving the p-Laplacian

We prove the existence of saddle solutions of nonlinear elliptic equation involving the p-Laplacian

On J. Schauder��s method to solve elliptic differential equations

The aim of this paper is to give complete proofs of the existence of classical C_2,a([`(W)]){C_{2,\alpha}\left(\bar{\Omega}\right)} solutions of linear and quasi-linear elliptic differential equations.     

Universal abelian covers of certain surface singularities

Every normal complex surface singularity with -homology sphere link has a universal abelian cover. It has been conjectured by Neumann and Wahl that the universal abelian cover of a rational or minimally elliptic singularity is a complete intersection singularity defined by a system of ``splice diagram equations'. In this paper we introduce a Neumann-Wahl system, which is an analogue of the system of splice diagram equations, and prove the following. If (X, o) is a rational or minimally elliptic singularity, then its universal abelian cover (Y, o) is an equisingular deformation of an isolated complete intersection singularity (Y₀, o) defined by a Neumann-Wahl system. Furthermore, if G denotes the Galois group of the covering Y → X, then G also acts on Y₀ and X is an equisingular deformation of the quotient Y₀/G. Dedicated to Professor Jonathan Wahl on his sixtieth birthday. This research was partially supported by the Grant-in-Aid for Young Scientists (B), The Ministry of Education, Culture, Sports, Science and Technology, Japan.     

The local and global parts of the basic zeta coefficient for operators on manifolds with boundary

For operators on a compact manifold X with boundary ∂X, the basic zeta coefficient C ₀(B, P _1,T ) is the regular value at s = 0 of the zeta function , where B = P ₊ + G is a pseudodifferential boundary operator (in the Boutet de Monvel calculus)—for example the solution operator of a classical elliptic problem—and P _1,T is a realization of an elliptic differential operator P ₁, having a ray free of eigenvalues. Relative formulas (e.g., for the difference between the constants with two different choices of P _1,T ) have been known for some time and are local. We here determine C ₀(B, P _1,T ) itself (with even-order P ₁), showing how it is put together of local residue-type integrals (generalizing the noncommutative residues of Wodzicki, Guillemin, Fedosov–Golse–Leichtnam–Schrohe) and global canonical trace-type integrals (generalizing the canonical trace of Kontsevich and Vishik, formed of Hadamard finite parts). Our formula generalizes a formula shown recently by Paycha and Scott for manifolds without boundary. It leads in particular to new definitions of noncommutative residues of expressions involving log P _1,T . Since the complex powers of P _1,T lie far outside the Boutet de Monvel calculus, the standard consideration of holomorphic families is not really useful here; instead we have developed a resolvent parametric method, where results from our calculus of parameter-dependent boundary operators can be used.     

Nonlocal fractional elliptic equations and applications

The L_p-coercive properties of a nonlocal fractional elliptic equation is studied. Particularly, it is proved that the fractional elliptic operator generated by this equation is sectorial in L_p space and also is a generator of an analytic semigroup. Moreover, by using the L_p-separability properties of the given elliptic operator the maximal regularity of the corresponding nonlocal fractional parabolic equation is established.     

Series expansions and maximum principles for generalized analytic vectors in the space Rn

In a few preceding papers we discussed the existence and representation of solutions of a certain of linear elliptic systems of partial differential equations of first order in the space R_n. Here we construct a complete set of solutions of these systems and prove maximum principles.     

A Sharp Upper Bound for the Number of Spanning Trees of a Graph

Let G = (V,E) be a simple graph with n vertices, e edges and d₁ be the highest degree. Further let λ_i, i = 1,2,...,n be the non-increasing eigenvalues of the Laplacian matrix of the graph G. In this paper, we obtain the following result: For connected graph G, λ₂ = λ₃ = ... =  λ_n-1 if and only if G is a complete graph or a star graph or a (d₁,d₁) complete bipartite graph. Also we establish the following upper bound for the number of spanning trees of G on n, e and d₁ only:

Local–global divisibility by 4 in elliptic curves defined over {\mathbb{Q}}

Let ${\mathcal{E}}Let E{\mathcal{E}} be an elliptic curve defined over \mathbbQ{\mathbb{Q}} . Let P ? E(\mathbb Q){P\in {\mathcal{E}}(\mathbb {Q})} and let q be a positive integer. Assume that for almost all valuations v ? \mathbbQ{v\in \mathbb{Q}} , there exist points D_v ? E(\mathbb Q_v){D_v\in {\mathcal{E}}(\mathbb {Q}_v)} such that P = qD _v . Is it possible to conclude that there exists a point D ? E(\mathbb Q){D\in {\mathcal{E}}(\mathbb {Q})} such that P = qD? A full answer to this question is known when q is a power of almost all primes p ? \mathbbN{p\in \mathbb{N}} , but some cases remain open when p ? S={2,3,5,7,11,13,17,19,37,43,67,163}{p\in S=\{2,3,5,7,11,13,17,19,37,43,67,163\}} . We now give a complete answer in the case when q = 4.     

On generalized modular forms supported on cuspidal and elliptic points

Suppose N∈{13,17,19,21,26,29,31,34,39,41,49,50}. In this paper, we extend previous results of Kohnen–Mason (On the canonical decomposition of generalized modular functions, 2010) to prove that generalized modular forms for Γ ₀(N) with rational Fourier expansions whose divisors are supported only at the cusps and at the elliptic points are actually classical modular forms. We discuss possible limitations to this extension and pose questions about possible zeroes for modular forms of prime level.     

Boundary blow-up solutions of p-Laplacian elliptic equations with lower order terms

We study the existence, uniqueness, and asymptotic behavior of blow-up solutions for a general quasilinear elliptic equation of the type −Δ _p u = a(x)u ^m −b(x)f(u) with p >  1 and 0 <  m <  p−1. The main technical tool is a new comparison principle that enables us to extend arguments for semilinear equations to quasilinear ones. Indeed, this paper is an attempt to generalize all available results for the semilinear case with p =  2 to the quasilinear case with p >  1.     

Regular boundary value problems for complete second order elliptic differential-operator equations in UMD Banach spaces

We consider coerciveness and Fredholmness of nonlocal boundary value problems for complete second order elliptic differential-operator equations in UMD Banach spaces. In some special cases, the main coefficients of the boundary conditions may be bounded operators and not only complex numbers. Then, we prove an isomorphism, in particular, maximal L _p -regularity, of the problem with a linear parameter in the equation. In both cases, the boundary conditions may also contain unbounded operators in perturbation terms. Finally, application to regular nonlocal boundary value problems for elliptic equations of the second order in non-smooth domains are presented. Equations and boundary conditions may contain differential-integral parts. The spaces of solvability are Sobolev type spaces W _p,q ^2,2. The first author is a member of G.N.A.M.P.A. and the paper fits the 60% research program of G.N.A.M.P.A.-I.N.D.A.M.; The third author was supported by the Israel Ministry of Absorption.     

Gaussian upper bounds for fundamental solutions of a family of parabolic equations

We study the family of divergence-type second-order parabolic equations w_e(x)\frac?u?t=div(a(x)w_e(x) ?u), x ? \mathbbRⁿ{\omega_\varepsilon(x)\frac{\partial u}{\partial t}={\rm div}(a(x)\omega_\varepsilon(x) \nabla u), x \in \mathbb{R}^n} , with parameter ${\varepsilon >0 }${\varepsilon >0 } , where a(x) is uniformly elliptic matrix and w_e=1{\omega_\varepsilon=1} for x _n  < 0 and w_e=e{\omega_\varepsilon=\varepsilon} for x _n  > 0. We show that the fundamental solution obeys the Gaussian upper bound uniformly with respect to e{\varepsilon} .     

The Klein correspondence of line congruences immersed in an elliptic spaceS 3 onto the hyperquadricP 4

This article is devoted to the investigation and the construction of the Klein correspondence of line congruences referred to a specialized moving frame in a 3-dimensional elliptic spaceS ₃ to the hyperquadricP ⁴ of the Klein 5-dimensional elliptic spaceS ₅. The Klein correspondence is given and characterized by Theorems 1, 2. The methods adapted here are based on Cartan's differential calculus [1], [6].     

On the rank of the fibers of elliptic K3 surfaces

Let X be an elliptic K3 surface endowed with two distinct Jacobian elliptic fibrations π _i , i = 1, 2, defined over a number field k. We prove that there is an elliptic curve C ⊂ X such that the generic rank over k of X after a base extension by C is strictly larger than the generic rank of X. Moreover, if the generic rank of π _j is positive then there are infinitely many fibers of π _i (j ≠ i) with rank at least the generic rank of π _i plus one.     

Poincaré Series of the Weyl Groups of the Elliptic Root Systems A 1 (1,1), A 1 (1,1)* and A 2 (1,1)

We calculate the Poincaré series of the elliptic Weyl group W(A ₂ ^(1,1)), which is the Weyl group of the elliptic root system of type A ₂ ^(1,1). The generators and relations of W(A ₂ ^(1,1)) have been already given by K. Saito and the author.