Analogues of hadwiger's theorem in space ℝ n —Sufficient conditions for a convex domain to enclose another

We estimate the kinematic measure of one convex domain moving to another under the groupG of rigid motions in ⁿ . We first estimate the kinematic formula for the total scalar curvature D ₀gD ₁ Rdv of then–2 dimensional intersection submanifold D ₀gD ₁. Then we use Chern and Yen's kinematic fundamental formula and our integral inequality to obtain a sufficient condition for one convex domain to contain another in ⁿ (4). Forn=4, we directly obtain another sufficient condition in ⁴.     

The Skorohod oblique reflection problem in domains with corners and application to stochastic differential equations

Summary The Skorohod oblique reflection problem for (D, , w) (D a general domain in ^d , (x),xD, a convex cone of directions of reflection,w a function inD(⁺, ^d )) is considered. It is first proved, under a condition on (D, ), corresponding to (x) not being simultaneously too large and too much skewed with respect to D, that given a sequence {w ⁿ} of functions converging in the Skorohod topology tow, any sequence {(x ⁿ, ⁿ)} of solutions to the Skorohod problem for (D, , w ⁿ) is relatively compact and any of its limit points is a solution to the Skorohod problem for (D, , w). Next it is shown that if (D, ) satisfies the uniform exterior sphere condition and another requirement, then solutions to the Skorohod problem for (D, , w) exist for everywD(⁺, ^d ) with small enough jump size. The requirement is met in the case when D is piecewiseC _b ¹ , is generated by continuous vector fields on the faces ofD and (x) makes and angle (in a suitable sense) of less than /2 with the cone of inward normals atD, for everyxD. Existence of obliquely reflecting Brownian motion and of weak solutions to stochastic differential equations with oblique reflection boundary conditions is derived.     

Existence theorems for nonlinear evolution inclusions

Summary In this paper we obtain an existence theorem for the abstract Cauchy problem for multivalued differential equations of the form u–^– f(u)+G(u), u(O)=x₀, where ^– f is the Fréchet subdifferential of a functionf defined on an open subset of a real separable Hilbert space H, taking its values in R {+} and G is a multifunction from C([0, T], ) into the nonempty subsets of L²([0, T], H). As an application we obtain an existence theorem for the multivalued perturbed problem x–^– f(x)+F(t, x), x(0)=x₀, where F:[0, T]×(H) is a multifunction satisfying some regularity assumptions.     

On monotone extensions of boundary data

Summary A functionf C (), is called monotone on if for anyx, y the relation x – y ₊ ^s impliesf(x)f(y). Given a domain with a continuous boundary and given any monotone functionf on we are concerned with the existence and regularity ofmonotone extensions i.e., of functionsF which are monotone on all of and agree withf on . In particular, we show that there is no linear mapping that is capable of producing a monotone extension to arbitrarily given monotone boundary data. Three nonlinear methods for constructing monotone extensions are then presented. Two of these constructions, however, have the common drawback that regardless of how smooth the boundary data may be, the resulting extensions will, in general, only be Lipschitz continuous. This leads us to consider a third and more involved monotonicity preserving extension scheme to prove that, when is the unit square [0, 1]² in ², strictly monotone analytic boundary data admit a monotone analytic extension.Research supported by NSF Grant 8922154Research supported by DARPA: AFOSR #90-0323     

Die Zahl der Spitzen und die Jacobi-Algebra einer isolierten Hyperflächensingularität

Let f{x_o,...,x_n} define a germ of a complex analytic hypersurface (X_o,0) with isolated singularity. We show that the number of cusps of the unfolded discriminant curve is an invariant of the Jacobian algebra {x,_o},...,x_n/(f/x_o,...,f/x_n) of f. Moreover we show that this number + 1 equals the sum of the Milnor numbers of (X_o,0) and of the polar curve of (X_o,0). Our result generalizes formulas of Iversen and Lê for plane curves to arbitrary dimensions.     

On improved regularity of weak solutions of some degenerate,Anisotropic elliptic systems

Summary We consider a (possibly) vector-valued function u: R^N, Rⁿ, minimizing the integral , 2-2/(n*1)<p<2, whereD _i u=u/x _i or some more general functional retaining the same behaviour, we prove higher integrability for Du: D₁ u,..., D_n–1 u L^p/(p-1) and D_nu L²; this result allows us to get existence of second weak derivatives: D(D₁ u),...,D(D_n–1u)L² and D(D_n u) L _p.This work has been supported by MURST and GNAFA-CNR.     

Sukzessive Minima mit und ohne Nebenbedingungen

Given semi-normsf andg on ⁿ and a real number >0. Then the successive minima off under the constraintg are defined by _j : = inf {: there existj linear independent vectors inZ ⁿ withf andg}. The main theorem of this paper (Lagrange multiplier theorem) states that the successive minima of a certainnorm h on ⁿ (without constraints) coincide with the _j 's up to bounded factors. Moreover, this norm is constructed explicitly. Using Minkowski's wellknown theorem on successive minima and our result certain inequalities on simultaneous Diophantine approximations are derived.     

The Darboux-Type Theorems for ∂-Symplectic and ∂-Contact Structures

A -symplectic structure on a complex manifold M of complex dimension2n is given by a smooth -closed (2, 0)-form such that ⁿ is nonvanishing. We prove that a version of the Darboux theorem isvalid for such a structure: locally can be represented as _i=1 ⁿ f _i f _{n + i} for appropriate smooth complex valuedfunctions f ₁, ..., f _2n . We also present a contact version of this theorem.     

On the Construction of Cosine Operator Functions and Semigroups on Function Spaces with Generator a(x)(d2dx2)+b(x)(d/dx)+c(x); Theory

In this paper we develop a method to solve exactly partial differential equations of the type ( ⁿ /t ⁿ )f(x,t)=(a(x)( ⁿ /x ⁿ )+b(x) (/x+c(x))f(x,t); n=1,2, with several boundary conditions, where f·,t) lies in a function space. The most powerful tool here is the theory of cosine operator functions and their connection to (holomorphic) semigroups. The method is that generally we are able to unify and generalize many theorems concerning problems in the theories of holomorphic semigroups, cosine operator functions, and approximation theory, especially these dealing with approximation by projections. These applications will be found in [14].     

Stabilization of solutions of certain parabolic equations and systems

This paper concerns the investigation of the stabilization of solutions of the Cauchy problem for a system of equations of the form u/t = u + f_i(u, v); v/t = v + F₂(u, v). It is proved that under certain assumptions the behavior of solutions as t is determined by mutual arrangement of the set of initial conditions {(u, v): u = f₁(x), v =f ₂(x), xRⁿ} and the trajectories of the system of ordinary differential equations du/dt = F₁(u, v), dv/dt = F₂(u, v). The question of stabilization of the solutions of a single quasilinear parabolic equation is also considered.Translated from Matematicheskie Zametki, Vol. 3, No. 1, pp. 85–92, January, 1968.     

Numerical implementation of coherence for the example of the “Swiss Cross”

Summary We consider the two-dimensional Helmholtz equation u+u=0 inD with the boundary conditionsu=0 on D. D is the Swiss Cross — a region consisting of five unit squares. A method based on the concept of Coherence is utilized to determine an approximation for the first eigenvalue= ₁ more accurate than calculated by classical difference methods. The numerical result is used to illustrate isoperimetric upper and lower bounds for ₁, and to test some conjectures on its relations with torsional rigidity.Dedicated to the memory of Professor Lathar Collatz     

Positive Solutions of Nonlinear Elliptic Equations in Unbounded Domains in R 2

We study the existence of positive solutions of the nonlinear equation u+f(,u)=0, in D with u=0 on D, where D is an unbounded domain in R ² with a compact nonempty boundary D consisting of finitely many Jordan curves. The aim is to prove an existence result for the above equation in a general setting by using potential theory.     

Behavior of the Bergman Kernel Near Smooth Convex Boundary Points

The nontangential behavior of the Bergman metric near a smooth convex boundary point of a bounded pseudoconvex domain D ⁿ is studied in terms of its multitype.Received January 25, 2002; in revised form June 20, 2002 Published online November 18, 2002     

The role of Fourier modes in extension theorems of Hartogs-Chirka type

We generalize Chirkas theorem on the extension of functions holomorphic in a neighbourhood of (F)(D×D) – where D is the open unit disc and (F) is the graph of a continuous D-valued function F – to the bidisc. We extend holomorphic functions by applying the Kontinuitätssatz to certain continuous families of analytic annuli, which is a procedure suited to configurations not covered by Chirkas theorem.Mathematics Subject classification (2000): 32D15Work supported by NSF Grant DMS-0072237.     

On the Uniqueness ofL 2-solutions in half-space of certain differential equations

Square integrable solutions to the equation{– ²/y² + P(D_x)+b(y)–}u(x, y) = f(x, y) are considered in the half-spacey>0, x ⁿ , whereP(D _x) is a constant coefficient operator. Under suitable conditions on lim_y0u(x, y), b(y), f(x, y) and , it is shown that suppu = suppf. This generalizes a result due to Walter Littman.Research partially supported by USNSF Grant 79-02538-A02.     

Contraction of Exterior Powers in Characteristic 2 and the Spin Module

Let K be a field of characteristic 2 and letV be a vector space of dimension 2m over K. Let f be a non-degenerate alternating bilinear form defined on V × V. The symplectic group Sp(2m, K) acts on the exterior powers ^k V for 0 k. 2m There is a contraction map defined on the exterior algebra , which commutes with the Sp(2m, K) action and satisfies ² = 0 and ( ^k V) ^k–1 V We prove that ( ^k V)= ker ^k–1 V except when k=m+2. In the exceptional case, ( ^m+2 V) has codimension 2^m in ker ^m V and we show that the quotient module ker ^m V/ ^m+2 V is a spin module for Sp(2m,K). When K is algebraically closed, we show that this spin module occurs with multiplicity 1 in ^m V and multiplicity 0 in all other components of V.     

A characterization of brownian motion in a lipschitz domain by its killing distributions

Let (Y _t, Q^x) be a strong Markov process in a bounded Lipschitz domainD with continuous paths up to its lifetime , and let (X _t, P^x) be a Brownian motion inD. IfY _– exists in D andQ ^x(Y_– C)=P^x(X_– C) for all Borel subsetsC of D and allx, thenY is a time change ofX.     

Die Singularitätenfunktionen der gespannten Platte und der Kreiszylinderschale

Zusammenfassung Es wird eine Darstellung einer Fundamentallösung des OperatorsP()=Q()²–(c₁)^2m durch Fundamentallösungen der OperatorenQ()±(c₁) ^m angegeben. Als Anwendung berechnen wir die Singularitätenfunktionen der gespannten Platte und der Kreiszylinderschale.

---

Summary A method is given, which allows to derive a fundamental solution of the operatorP()=Q()²–(c₁)^2m from some fundamental solutions of the operatorsQ()±(c₁) ^m . As an application we easily obtain the singular solutions of the unidirectionally stretched plate and of the circular cylindrical shell.

     

Local partial regularity theorems for suitable weak solutions of a class of degenerate systems

A partial regularity theorem is established for a particular class of weak solutions to the systemu/t– div(K(u)u)=(u)¦¦², div((u))=0 on a bounded domain inR ^N . Under our assumptions, (u) may exhibit exponential decay, and thus the system may be degenerate. Our proof is based upon a blow-up argument.This work was supported in part by NSF Grant DMS9424448.     

Projections on Lp-spaces of polyanalytic functions

The fundamental result: for an arbitrary bounded, simply connected domain in , the subspace L_n,m ^p() of the space L^p(, ) ( is the plane Lebesgue measure, p 1), consisting of the (m, n)-analytic functions in , is complemented in LP(, ) (a function f is said to be (m, n)-analytic if (^m+n/¯Z^mZⁿ)f=0 in ). Consequently, by virtue of a theorem of J. Lindenstrauss and A. Pelczyski, the space L_n,m ^P() is linearly homeomorphic to lP. In particular, for m=n=1 we obtain that the space of all harmonic L^P-functions in is complemented in L^P(, ). This result has been known earlier only for smooth domains.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 190, pp. 15–33, 1991.