The Zaremba Problem with Singular Interfaces as a Corner Boundary Value Problem

We study mixed boundary value problems for an elliptic operator on a manifold with boundary , i.e., in on , where is subdivided into subsets with an interface and boundary conditions on that are Shapiro–Lopatinskij elliptic up to from the respective sides. We assume that is a manifold with conical singularity . As an example we consider the Zaremba problem, where is the Laplacian and Dirichlet, Neumann conditions. The problem is treated as a corner boundary value problem near which is the new point and the main difficulty in this paper. Outside the problem belongs to the edge calculus as is shown in Bull. Sci. Math. (to appear).With a mixed problem we associate Fredholm operators in weighted corner Sobolev spaces with double weights, under suitable edge conditions along of trace and potential type. We construct parametrices within the calculus and establish the regularity of solutions.     

Two-parameter p, q-variation Paths and Integrations of Local Times

In this paper, we prove two main results. The first one is to give a new condition for the existence of two-parameter -variation path integrals. Our condition of locally bounded -variation is more natural and easy to verify than those of Young. This result can be easily generalized to multi-parameter case. The second result is to define the integral of local time pathwise and then give generalized It’s formula when is only of bounded -variation in . In the case that is of locally bounded variation in , the integral is the Lebesgue–Stieltjes integral and was used by Elworthy, Truman and Zhao. When is of only locally -variation, where , , and , the integral is a two-parameter Young integral of -variation rather than a Lebesgue–Stieltjes integral. In the special case that is independent of , we give a new condition for Meyer's formula and is defined pathwise as a Young integral. For this we prove the local time is of -variation in for each , for each almost surely (-variation in the sense of Lyons and Young, i.e. ).     

Existence and Stability Results for Renormalized Solutions to Noncoercive Nonlinear Elliptic Equations with Measure Data

In this paper we prove the existence of a renormalized solution to a class of nonlinear elliptic problems whose prototype is

Regularity in Capacity and the Dirichlet Laplacian

Given an open set in , we prove that every function in is zero everywhere on the boundary if and only if is regular in capacity. If in addition is bounded, then it is regular in capacity if and only if the mapping from into is injective, where denotes the Perron solution of the Dirichlet problem. Let be the set of all open subsets of which are regular in capacity. Then one can define metrics and on only involving the resolvent of the Dirichlet Laplacian. Convergence in those metrics will be defined to be the local/global uniform convergence of the resolvent of the Dirichlet Laplacian applied to the constant function . We prove that the spaces and are complete and contain the set of all open sets which are regular in the sense of Wiener (or Dirichlet regular) as a closed subset.     

Uniform Estimates of the Fundamental Solution for a Family of Hypoelliptic Operators

In this paper we are concerned with a family of elliptic operators represented as sum of square vector fields: , in where is the Laplace operator, , and the limit operator is hypoelliptic. It is well known that admits a fundamental solution . Here we establish some a priori estimates uniform in of it, using a modification of the lifting technique of Rothschild and Stein. As a consequence we deduce some a priori estimates uniform in , for solutions of the approximated equation . These estimates can be used in particular while studying regularity of viscosity solutions of nonlinear equations represented in terms of vector fields.     

Intrinsic Ultracontractivity for Schrödinger Operators with Mixed Boundary Conditions

Let be a domain in , . Let be a divergence form uniformly elliptic operator with Dirichlet boundary conditions on and Neumann boundary conditions on , where is a closed subset of . We prove intrinsic ultracontractivity for the semigroup associated to the Schrödinger operator , where is a potential in the Kato class, provided that is locally Lipschitz and is given by the boundary of either a Hölder domain of order or a uniformly Hölder domain of order , . Our results extend to the mixed boundary case the results of Bañuelos, Bass and Burdzy, Bass and Hsu, and Davies and Simon.     

Young Integrals and SPDEs

In this note, we study the non-linear evolution problem $dY_t = -A Y_t dt + B(Y_t) dX_t,$ where $XIn this note, we study the non-linear evolution problem where is a -H?lder continuous function of the time parameter, with values in a distribution space, and the generator of an analytical semigroup. Then, we will give some sharp conditions on in order to solve the above equation in a function space, first in the linear case (for any value of in ), and then when satisfies some Lipschitz type conditions (for ). The solution of the evolution problem will be understood in the mild sense, and the integrals involved in that definition will be of Young type.     

Symmetric Stable Processes on Unbounded Domains

Let be a symmetric -stable process in , , . We give necessary and sufficient condition under which the expectation of a very general function of the exit time from horns is finite. These domains include the symmetric domains given by increasing functions studied earlier by various authors. Our methods differ from those in earlier papers in that we obtain our results from estimates on the transition densities instead of harmonic measure. Some of this estimates are of independent interest.Supported in part by NSF grant #9700585-DMS and RTN Harmonic Analysis and Related Problems contract HPRN-CT-2001-00273-HARP.     

Quasi-regular Dirichlet Forms and L^p-resolvents on Measurable Spaces

We prove that for any semi-Dirichlet form on a measurable Lusin space E there exists a Lusin topology with the given -algebra as the Borel -algebra so that becomes quasi-regular. However one has to enlarge E by a zero set. More generally a corresponding result for arbitrary -resolvents is proven.     

On Auslander–Reiten Components and Heller Lattices for Integral Group Rings

Let be a finite group, a complete discrete valuation ring of characteristic zero with residue class field of characteristic , and a block of the group ring . Suppose that is of infinite representation type and is sufficiently large to satisfy certain conditions. Let be the Auslander–Reiten quiver of and a connected component of . In this paper, we show that if contains some Heller lattices then the tree class of the stable part of is . Also, we show that has infinitely many components of type if a defect group of is neither cyclic nor a Klein four group.Presented by Jon Carlson.     

Deep Matrix Algebras of Finite Type

Deep matrix algebras based on a set over a ring are introduced and studied by McCrimmon when has infinite cardinality. Here, we construct a new -module that is faithful for of any cardinality. For a field of arbitrary characteristic and , is studied in depth. The algebra is shown to possess a unique proper non-zero ideal, isomorphic to . This leads to a new and interesting simple algebra, , the quotient of by its unique ideal. Both and the quotient algebra are shown to have centers isomorphic to . Via the new faithful representation, all automorphisms of are shown to be inner in the sense of Definition 18.Presented by D. Passman.     

Admissible Galois Structures and Coverings in Regular Mal’cev Categories

Given a regular Gumm category such that any regular epimorphism is effective for descent, we prove that any Birkhoff subcategory in gives rise to an admissible Galois structure. This result allows one to consider some new applications of the categorical Galois theory in the context of topological algebras. Given a regular Mal’cev category , we first characterize the coverings of the Galois structure induced by the subcategory of the abelian objects in . Then we consider as a subcategory of the category of the equivalence relations in , and we characterize the coverings of the corresponding Galois structure . By composing the Galois structures and we obtain the Galois structure induced by as a subcategory of . We give the characterization of the -coverings in terms of the coverings of and .     

Second Order Elliptic Equations in $\mathbb{R}^{d} $ with Piecewise Continuous Coefficients

The existence and uniqueness of solutions of second order elliptic differential equations in are proved. The coefficients of second order terms are allowed to have discontinuity at finitely many parallel hyper-planes in and the first derivatives of solutions can have jumps at the hyper-planes.      

Involutions of \operatorname{{SL}} (nk), (n>2)

A first characterization of the isomorphism classes of -involutions for any reductive algebraic group defined over a perfect field was given in [7] using three invariants. In this paper we give a simple characterization of the isomorphism classes of involutions of with any field of characteristic not equal to . We classify the isomorphism classes of involutions for algebraically closed, the real numbers, the -adic numbers and finite fields. We also determine in which cases the corresponding fixed point group is -anisotropic. In those cases the corresponding symmetric -variety consists of semisimple elements.Aloysius G. Helminck was partially supported by N.S.F. Grant DMS-9977392.     

Universality of Coproducts in Categories of Lax Algebras

Categories of lax -algebras are shown to have pullback-stable coproducts if preserves inverse images. The general result not only gives a common proof of this property in many topological categories but also shows that important topological categories, like the category of uniform spaces, are not presentable as a category of lax -algebras, with preserving inverse images. Moreover, we show that any such category of -algebras has a concrete, coproduct preserving functor into the category of topological spaces.     

Riesz and Poisson-Jensen Representation Formulas for a Class of Ultraparabolic Operators on Lie Groups

The aim of this paper is to give some representation formulas of Riesz and Poisson-Jensen type for super-solutions to a class of hypoelliptic ultraparabolic operators on a homogeneous Lie group . Our results complete the ones obtained in Cinti (Math Scand 100:1–21, 2007). We also provide a suitable theory for -Green functions and for -Green potentials of Radon measures. The proofs mostly rely on the use of appropriate techniques relevant to the Potential Theory for . Investigation supported by University of Bologna. Funds for selected research topics.     

A representation of bivariate extreme value distributions via norms on \mathbb{R}^{2}

It is known that a bivariate extreme value distribution (EVD) with reverse exponential margins can be represented as , , where is a suitable norm on . We prove in this paper the converse implication, i.e., given an arbitrary norm on , , , defines an EVD with reverse exponential margins, if and only if the norm satisfies for the condition . This result is extended to bivariate EVDs with arbitrary margins as well as to extreme value copulas. By identifying an EVD , , with the unit ball corresponding to the generating norm , we obtain a characterization of the class of EVDs in terms of compact and convex subsets of .     

Descending Chains and Antichains of the Unary, Linear, and Monotone Subfunction Relations

The -subfunction relations on the set of operations on a finite base set defined by function classes are examined. For certain clones on , it is determined whether the partial orders induced by the respective -subfunction relations have infinite descending chains or infinite antichains. More specifically, we investigate the subfunction relations defined by the clone of all functions on , the clones of essentially at most unary operations, the clones of linear functions on a finite field, and the clones of monotone functions with respect to the various partial orders on .     

The Pluri-Fine Topology is Locally Connected

We prove that the pluri-fine topology on any open set in is locally connected. This answers a question by Fuglede.     

Epi-topology and Epi-convergence for Archimedean Lattice-ordered Groups with Unit

is the category of archimedean -groups with distinguished weak order unit, with -group homomorphisms which preserve unit. This category includes all rings of continuous functions and all rings of measurable functions modulo null functions, with ring homomorphisms. The authors, and others, have studied previously the epimorphisms (right-cancellable morphisms) in . There is a rich theory. In this paper, we describe a topological approach to the analysis of these epimorphisms. On each – object, we define a topology and a convergence . These have the same closure operator, and this closure “captures epics” in the sense: a divisible subobject of is dense iff is epically embedded. The topology is , but only sometimes Hausdorff or an -group topology. The convergence is a Hausdorff -group convergence, but only sometimes topological. The associations of to , and to , are functorial. Dedicated to Bernhard Banaschewski for his 80th birthday.