Subelliptic Cordes estimates

We consider second-order linear parabolic operators in non-divergence form that are intrinsically defined on Riemannian manifolds. In the elliptic case, Cabr\'e proved a global Krylov-Safonov Harnack inequality under the assumption that the sectional curvature of the underlying manifold is nonnegative. Later, Kim improved Cabr\'e's result by replacing the curvature condition by a certain condition on the distance function. Assuming essentially the same condition introduced by Kim, we establish Krylov-Safonov Harnack inequality for nonnegative solutions of the non-divergent parabolic equation. This, in particular, gives a new proof for Li-Yau Harnack inequality for positive solutions to the heat equation in a manifold with nonnegative Ricci curvature.

On every right circular cone and for every uniformly elliptic operator in nondivergence form there exists a barrier, that is, a supersolution which is zero at the vertex and positive elsewhere on the closed cone. These barriers are applied to the Dirichlet problem. Other applications are deferred to a later paper on « extremal barriers »

Let  ⇢ Rn be a bounded open set satisfying the uniform exterior cone condition. Let A be a uniformly elliptic operator given by Au = Xn i, j=1 ai j @i j u + Xn j=1 b j @ j u + cu where a j i = ai j 2 C(¯ ) and b j , c 2 L1(), c  0 . We show that the realization A0 of A in C0() := {u 2 C(¯ ) : u|@ = 0} given by D(A0) := {u 2 C0() W2,n loc () : Au 2 C0()} A0u := Au generates a bounded holomorphic C0-semigroup on C0(). The result is in particular true if  is a Lipschitz domain. So far the best known result seems to be the case where  has C2-boundary [12, Section 3.1.5]. We also study the elliptic problem −Au = f u|@ = g .

We give a detailed analysis of supersolutions of form rλf(θ) for the classLα of uniformly elliptic operators in nondivergence form with ellipticity constant α. Fundamental to the analysis is the extremal supersolution rλFλ, α(θ) for each real λ, which is itself a solution of a certain extremal elliptic equation and which remains positive on a cone of wider aperature than any other supersolution of this form. These supersolutions are employed as “barriers” to yeild Phragmen-Lindelof theorems (λ>0) for unbounded domains contained in a cone, and Holder continuity (λ>0) and removeable singularity (λ<0) results at boundary points with an exterior cone property. In each case the growth condition 0(rλ) involved, depending on the aperature of the cone, is best possible. Similar results carry through for operators with singular lower order terms and possibily positive zero order coefficient.

We consider non-negative solutions to a class of second-order degenerate Kolmogorov operators of the form L=Sigma(m)(i,j=1)a(i,j)(z)partial derivative(xixj)+Sigma(m)(i=1)a(i)(z)partial derivative(x ...

Potential analysis for a class of diffusion equations: A Gaussian bounds approach

We consider two algorithmic problems concerning sub-semigroups of Heisenberg groups and, more generally, two-step nilpotent groups. The first problem is Intersection Emptiness, which asks whether a finite number of given finitely generated semigroups have empty intersection. This problem was first studied by Markov in the 1940s. We show that Intersection Emptiness is PTIME decidable in the Heisenberg groups H_n(𝕂) over any algebraic number field 𝕂, as well as in direct products of Heisenberg groups. We also extend our decidability result to arbitrary finitely generated 2-step nilpotent groups. The second problem is Orbit Intersection, which asks whether the orbits of two matrices under multiplication by two semigroups intersect with each other. This problem was first studied by Babai et al. (1996), who showed its decidability within commutative matrix groups. We show that Orbit Intersection is decidable within the Heisenberg group H₃(ℚ).

The [formula omitted] Dirichlet problem for second-order, non-divergence form operators: solvability and perturbation results

We consider the Fuchsian linear differential equation obtained (modulo a prime) for , the five-particle contribution to the susceptibility of the square lattice Ising model. We show that one can understand the factorization of the corresponding linear differential operator from calculations using just a single prime. A particular linear combination of and can be removed from and the resulting series is annihilated by a high order globally nilpotent linear ODE. The corresponding (minimal order) linear differential operator, of order 29, splits into factors of small orders. A fifth-order linear differential operator occurs as the left-most factor of the ‘depleted’ differential operator and it is shown to be equivalent to the symmetric fourth power of LE, the linear differential operator corresponding to the elliptic integral E. This result generalizes what we have found for the lower order terms and . We conjecture that a linear differential operator equivalent to a symmetric (n − 1) th power of LE occurs as a left-most factor in the minimal order linear differential operators for all 's.

We use transfinite induction to show that if L is an epimorphism of a vector space V and maps a vector subspace W of V into a proper subspace of itself, then there is a smallest subspace E of V containing W such that L(E) E (or a minimal space of surjectivity or solvability) and we give examples where there are infinitely many distinct minimal spaces of solvability. We produce an example showing that if L 1 and L 2 are two epimorphisms of a vector space V which are endomorphisms of a proper subspace W of V such that LI(W) n L2(W is a proper subspace of W, then there may not exist a smallest subspace E of V containing W such that LI(E) E L2(E). While no nonconstant linear partial differential operator maps the field of meromorphic functions onto itself, we construct a locally convex topological vector space of formal power series containing the meromorphic functions such that every linear partial differential operator with constant coefficients maps this space linearly and continuously onto itself. Furthermore, we show that algebraically there is for every linear partial differential operator P(D) with constant coefficients a smallest extension E of the meromorphic functions in n complex variables, where

On contact manifolds we describe a notion of (contact) finite-type for linear partial differential operators satisfying a natural condition on their leading terms. A large class of linear differential operators are of finite-type in this sense but are not well understood by currently available techniques. We resolve this in the following sense. For any such D we construct a partial connection r H on a (finite rank) vector bundle with the property that sections in the null space of D correspond bijectively, and via an explicit map, with sections parallel for the partial connection. It follows that the solution space of D is finite dimensional and bounded by the corank of the holonomy algebra of r H. The treatment is via a uniform procedure, even though in most cases no normal Cartan connection is available. The prolongations of a k th order linear differential operator between vector bundles arise by differentiating the given operator D : E ! F, and forming a new system comprising D along with auxiliary operators that capture some of this derived data. To exploit this effectively it is crucial to determine what part of this information should be retained, and then how best to manage it. With this understood, for many classes of operators the resulting prolonged operator can expose key properties of the original differential operator and its equation. Motivated by questions related to integrability and deformations of structure, a theory of overdetermined equations and prolonged systems was developed during the 1950s and 1960s by Goldschmidt, Spencer, and others (2, 17). Generally, results in these works are derived abstractly using jet bundle theory, and are severely restricted in the sense that they apply most readily to differential operators satisfying involutivity conditions. These features mean the theory can be difficult to apply. In the case that the given partial differential operator D : E ! F, has surjective symbol there is an effective algorithmic approach to this problem. The prolongations are constructed from the leading symbol �(D) : J k � 1 E ! F, where J k � 1 is the bundle of symmetric covariant tensors on M of rank k. At a point of M, denoting by K the kernel of �(D), the spaces K ` = ( J ` � 1 K) ( J k+` � 1 E), `� 0, capture spaces of new variables to be introduced, and the system closes up if K ` = 0 for

This article is a sequel to the earlier articles, which describe the invertible ordinary differential operators and their generalizations. The generalizations are invertible mappings of filtered modules generated by one differentiation, and are called invertible D-operators. In particular, invertible ordinary linear differential operators, invertible linear difference operators with periodic coefficients, maps defined by unimodular matrices, and C-transformations of control systems are invertible D-operators. C-Transformations are those invertible transformations for which the variables of one system are expressed in terms of the variables of the other system and their derivatives.In the article we consider the invertible D-operators whose inverses are D-operators of the same type. In previous papers, a classification of invertible D-operators was obtained. Namely, a table of integers was associated to each invertible D-operator. These tables were described in a clear elementary-geometric language. Thus, to each invertible D-operator one assigns an elementary-geometric model, which is called a d-scheme of squares. The class of invertible D-operators having the same d-scheme was also described. In this paper, the invertible D-operators whose d-schemes consist of a single square are called unicellular. It is proved that any unicellular operator in some bases is given by an upper triangular matrix that differs from the identity matrix only by the first row. The main result is representation of the arbitrary invertible D-operator as a composition of unicellular operators. The minimum number of unicellular operators in such a composition is equal to the number of squares of the d-scheme of the original D-operator. As in previous papers, the used method is based on the description of d-schemes in the language of spectral sequences of algebraic complexes.The results obtained can be useful in the transformation and classification of control systems, in particular to describe flat systems.

Analysis on the Heisenberg group has become an important area with strong connections to Fourier analysis, group representations, and partial differential operators. We propose to show in this work that special functions methods can also play a significant part in this theory. There is a one-parameter family of second-order hypoelliptic operators Lγ, (γ ∊ C), associated to the Laplacian L0 (also called the subelliptic or Kohn Laplacian). These operators are closely related to the unit ball for reasons of homogeneity and unitary group invariance. The associated Dirichlet problem is to find functions with specified boundary values and annihilated by Lγ inside the ball (that is, Lγ-harmonic). This is the topic of this paper.Gaveau [9] proved the first positive result, showing that continuous functions on the boundary can be extended to L0-harmonic functions in the ball, by use of diffusion-theoretic methods. Jerison [15] later gave another proof of the L0-result. Hueber [14] has recently obtained some results dealing with special values of the Poisson kernel for L0.

Let Ω be a domain in ℝ N and consider a second order linear partial differential operator A in divergence form on Ω which is not required to be uniformly elliptic and whose coefficients are allowed to be complex, unbounded and measurable. Under rather general conditions on the growth of the coefficients we construct a quasi-contractive analytic semigroup \((e^{-t A_{V}})_{t\geqslant0}\) on L 2(Ω,dx), whose generator A V gives an operator realization of A under general boundary conditions. Under suitable additional conditions on the imaginary parts of the diffusion coefficients, we prove that for a wide class of boundary conditions, the semigroup \((e^{-t A_{V}})_{t\geqslant0}\) is quasi-L p -contractive for 1<p<∞. Similar results hold for second order nondivergence form operators whose coefficients satisfy conditions similar to those on the coefficients of the operator A, except for some further requirements on the diffusion coefficients. Some examples where our results can be applied are provided.

In this paper a generalized approach to several universality results is given by replacing holomorphic or harmonic functions by zero solutions of arbitrary linear partial differential operators. Instead of the approximation theorems of Runge and others, we use an approximation theorem of Hormander.