Oblique derivative problems for second order nonlinear equations of mixed type in multiply connected domains

In this paper, the unique solvability of oblique derivative boundary value problems for second order nonlinear equations of mixed (elliptic-hyperbolic) type in multiply connected domains is proved, which mainly is based on the representation of solutions for the above boundary value problem, and the uniqueness and existence of solutions of the above problem for the equation u_xx + sgn y u_yy = 0.     

W 2,p -a priori estimates for the emergent Poincaré Problem

We derive W ^2,p (Ω)-a priori estimates with arbitrary p ∈(1, ∞), for the solutions of a degenerate oblique derivative problem for linear uniformly elliptic operators with low regular coefficients. The boundary operator is given in terms of directional derivative with respect to a vector field ℓ that is tangent to ∂Ω at the points of a non-empty set ε ⊂ ∂Ω and is of emergent type on ∂Ω.      

Free boundary and mixed boundary value problems for quasilinear elliptic equations: tangential oblique derivative and degenerate Dirichlet boundary problems

We present Hölder estimates and Hölder gradient estimates for a class of free boundary problems with tangential oblique derivative boundary conditions provided the oblique vector β does not vanish at any point on the boundary. We also establish the existence result for a general class of quasilinear degenerate problems of this type including nonlinear wave systems and the unsteady transonic small disturbance equation.     

Maximum estimates for oblique derivative problems with right hand side in L p , p<n

If u is a solution of the second order elliptic differential equation Lu=f in some Lipschitz domain in ⁿ , we estimate the maximum of u in terms of some oblique derivative prescribed on the boundary of the domain and in terms of the L ^p norm of f with p<n.     

On the existence of Feller semigroups with discontinuous coefficients II

This paper is devoted to the functional analytic approach to the problem of existence of Markov processes with Dirichlet boundary condition, oblique derivative boundary condition and first-order Wentzell boundary condition for second-order, uniformly elliptic differential operators with discontinuous coefficients. More precisely, we construct Feller semigroups associated with absorption, reflection, drift and sticking phenomena at the boundary. The approach here is distinguished by the extensive use of the ideas and techniques characteristic of the recent developments in the Calderon- Zygmund theory of singular integral operators with non-smooth kernels.     

Ein schiefes Randwertproblem zu einer elliptischen Differentialgleichung zweiter Ordnung mit einer expliziten L2-Abschätzung für die zweiten Ableitungen der Lösung

In connection with the free boundary value problem of determining the earth's surface from measurements of gravitational potential and force-field (“the geodetic boundary problem”), an oblique derivative problem arises, where D₀ is some bounded domain, star shaped with respect to the origin. In order to prove a uniquencess theorem for the geodetic boundary problem, it is essential to give estimates for (weighted) L₂-norms of the second derivatives of the solutions so that their bounds can be estimated numerically if bounds for the function describing the boundary are known. In this paper a Fredholm inverse for the above problem is constructed and the second derivatives of the solutions are estimated in the desired form.     

Smoothness of solutions of a special one-sided problem

We consider a planar domain, namely a curvilinear quadrilateral. We study a variational inequality of special form on the set of functions that are monotonically increasing on part of the boundary. This problem corresponds to a one-sided problem for an elliptic equation. A boundary condition of first kind is prescribed on part of the boundary, while on the other part of the boundary the tangential derivative is nonnegative and the product of the tangential and oblique derivatives is zero. We establish that the first derivatives of the solution satisfy a Hölder condition. Bibliography: 5 titles.Translated fromProblemy Matematicheskogo Analiza, No. 12, 1992, pp. 173–186.     

Schauder estimates for oblique derivative problems

We prove that the solution of the oblique derivative parabolic problem in a noncylindrical domain Ω^T belongs to the anisotropic Holder space C^{2+α, 1+α/2}(gwT) 0 < α < 1, even if the nonsmooth “lateral boundary” of Ω^T is only of class C^{1+α, (1+α)/2}). As a corollary, we also obtain an a priori estimate in the Hölder space C^2+α(Ω₀) for a solution of the oblique derivative elliptic problem in a domain Ω₀ whose boundary belongs only to the classe C^1+α.     

On the regularity of the oblique derivative problem for quasilinear elliptic systems

This note continues the author's investigations of the regularity problem for quasilinear elliptic systems with nonlinear boundary conditions. Partial regularity of weak solutions of the oblique derivative problem is proved here. Bibliography: 7 titles. Dedicated to V. A. Solonnikov on his sixtieth anniversary Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 213, 1994, pp. 5–13. Translated by S. Yu. Pilyugin.     

A counterexample in the theory of best approximation

We give an example of a domain Ω with smooth boundary and with compact subsets K₁ and K₂, such that K₁ and K₂ have disjoint hulls, but such that there is no function u, harmonic on Ω, which is negative on K₁ and positive on K₂.     

Index in K-Theory for Families of Fibred Cusp Operators

A families index theorem in K-theory is given for the setting of Atiyah, Patodi, and Singer of a family of Dirac operators with spectral boundary condition. This result is deduced from such a K-theory index theorem for the calculus of cusp, or more generally fibred-cusp, pseudodifferential operators on the fibres (with boundary) of a fibration; a version of Poincaré duality is also shown in this setting, identifying the stable Fredholm families with elements of a bivariant K-group. (Received: February 2006)     

Relative k-cycles and local elliptic boundary conditions

In this paper, we discuss the following conjecture raised by Baum and Douglas: For any first order elliptic differential operator D on a smooth manifold M with boundary ?M D possesses a (local) elliptic boundary condition if and only if ?[D]=0 in K₁(?M), where [D] is the relative K-cycle in K_o(M,?M) corresponding to D. We prove the “if” part of this conjecture for dim(M)≠4,5,6,7 and the “only if” part of the conjecture for arbitrary dimension.     

Model oblique derivative problem for the heat equation with a discontinuous boundary function

The oblique derivative problem for the heat equation is considered in a model formulation with a boundary function that can be discontinuous and with the boundary condition understood as the limit in the normal direction almost everywhere on the lateral boundary of the domain. An example is given showing that the solution is not unique in this formulation. A solution is sought in the parabolic Zygmund space H ₁, which is an analogue of the parabolic Hölder space for an integer smoothness exponent. A subspace of H ₁ is introduced in which the existence and uniqueness of the solution is proved under suitable assumptions about the data of the problem.     

A Kinematic Formula for Affine Diameters and Affine Medians of a Convex Set

For a planar convex set K with C ²-smooth boundary, the area of the set of the points lying on a given number of affine diameters of K is estimated. As a corollary, it is proved that the area of K is at most pM² /4\pi M^2 /4 , where M is the largest length of a chord of K halving the area of K. Bibliography: 2 titles.     

Optimality Criteria and Duality in Multiobjective Programming Involving Nonsmooth Invex Functions

In this paper a generalization of invexity is considered in a general form, by means of the concept of K-directional derivative. Then in the case of nonlinear multiobjective programming problems where the functions involved are nondifferentiable, we established sufficient optimality conditions without any convexity assumption of the K-directional derivative. Then we obtained some duality results.     

Hyperbolic manifolds with convex boundary

Let (M,∂M) be a 3-manifold, which carries a hyperbolic metric with convex boundary. We consider the hyperbolic metrics on M such that the boundary is smooth and strictly convex. We show that the induced metrics on the boundary are exactly the metrics with curvature K>-1, and that the third fundamental forms of ∂M are exactly the metrics with curvature K<1, for which the closed geodesics which are contractible in M have length L>2π. Each is obtained exactly once. Other related results describe existence and uniqueness properties for other boundary conditions, when the metric which is achieved on ∂M is a linear combination of the first, second and third fundamental forms.     

Model oblique derivative problem for the heat equation in the Zygmund space <Emphasis Type="Italic">H</Emphasis> <Subscript>1</Subscript>

The third boundary value problem and the oblique derivative problem for the heat equation are considered in model formulations. A difference compatibility condition is introduced for the initial and boundary functions. Under suitable assumptions made about the problem data, the solutions are shown to belong to the parabolic Zygmund space H₁, which is the analogue of the parabolic Hölder space for an integer smoothness exponent.     

Konstruktion regulärer Hüllen konstanter Breite

LetM be a compact, convex set of diameter 2 inE ^d. There exists a bodyK of constant width 2 containingM such that every symmetry ofM is one ofK and every singular boundary point ofK is a boundary point ofM, for which the set of antipodes inK is the convex hull of the antipodes, which are already inM.

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Solutions of model heat problems in Zygmund spaces

We consider the first boundary value problem and the oblique derivative problem for the heat equation in the model case where the domain is a half-layer and the coefficients of the boundary operator in the oblique derivative problem are constant. Under the corresponding assumptions on the problem data, we show that the solutions belong to anisotropic Zygmund spaces, which “close” the scale of anisotropic Hölder spaces for integer values of the smoothness exponent.