Existence and non-existence of solutions for a class of Monge-Ampère equations

We study the boundary value problems for Monge-Ampère equations: detD²u=e−u in Ω⊂Rn, n?1, u_|∂Ω=0. First we prove that any solution on the ball is radially symmetric by the argument of moving plane. Then we show there exists a critical radius such that if the radius of a ball is smaller than this critical value there exists a solution, and vice versa. Using the comparison between domains we can prove that this phenomenon occurs for every domain. Finally we consider an equivalent problem with a parameter detD²u=e−tu in Ω, u_|∂Ω=0, t?0. By using Lyapunov-Schmidt reduction method we get the local structure of the solutions near a degenerate point; by Leray-Schauder degree theory, a priori estimate and bifurcation theory we get the global structure.     

Construction of Carleman formulas by using mixed problems with parameter-dependent boundary conditions

Let D be an open connected subset of the complex plane C with sufficiently smooth boundary ?D. Perturbing the Cauchy problem for the Cauchy–Riemann system ??u = f in D with boundary data on a closed subset S ? ?D, we obtain a family of mixed problems of the Zaremba-type for the Laplace equation depending on a small parameter ε ∈ (0, 1] in the boundary condition. Despite the fact that the mixed problems include noncoercive boundary conditions on ?D\S, each of them has a unique solution in some appropriate Hilbert space H ⁺(D) densely embedded in the Lebesgue space L ₂(?D) and the Sobolev–Slobodetski? space H ^1/2?δ(D) for every δ > 0. The corresponding family of the solutions {u _ε} converges to a solution to the Cauchy problem in H ⁺(D) (if the latter exists). Moreover, the existence of a solution to the Cauchy problem in H ⁺(D) is equivalent to boundedness of the family {u _ε} in this space. Thus, we propose solvability conditions for the Cauchy problem and an effective method of constructing a solution in the form of Carleman-type formulas.     

Rapid zonal algorithm for polyelliptic PDEs in domains with high aspect ratio

Presented herein is a zonal boundary element method (ZBEM) for the rapid and efficient solution of a wide class of polyelliptic boundary value problems which can be recast in integral-equation form, in domains with high aspect ratio (L⪢ 1). In contrast to the dense-matrix solution procedure of the classical BEM (CBEM), the ZBEM employs a sparse, block-tridiagonal matrix solution technique which admits rapid inversion. Our large-L asymptotic theory predicts the ZBEM to be O(L²) times faster than, and require O(L⁻¹) times the storage of, the equivalent-resolution CBEM. By implementing the ZBEM on two engineering-based harmonic and biharmonic example boundary value problems, up to l = 1000, we are able to demonstrate excellent agreement between our numerical results and our asymptotic theory. We suggest that the ZBEM permits the economical solution of a wide class of problems which were hitherto resolvable on only the largest computational platforms.     

On approximation in the fictitious domain method for a bidimensional transport equation

In this article a numerical method for solving a two‐dimensional transport equation in the stationary case is presented. Using the techniques of the variational calculus, we find the approximate solution for a homogeneous boundary‐value problem that corresponds to a square domain D₂. Then, using the method of the fictitious domain, we extend our algorithm to a boundary value problem for a set D that has an arbitrary shape. In this approach, the initial computation domain D (called physical domain) is immersed in a square domain D₂. We prove that the solution obtained by this method is a good approximation of the exact solution. The theoretical results are verified with the help of a numerical example. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

Spectral properties of the neutron transport equation

The general equation describing the steady-state flow through a porous column is λu ? D_xA(D_x?(u) + G(u)) = f, where λ is a nonnegative constant. In this paper existence, uniqueness and comparison results for solutions to the Dirichlet and mixed boundary value problems associated with this equation are proven. The existence of a weak solution to the evolution problems associated with the equation u_t = D_x(D_x?(u) + G(u)) are deduced.     

The class of solenoidal planar-helical vector fields

The class of solenoidal vector fields whose lines lie in planes parallel to R ² is constructed by the method of mappings. This class exhausts the set of all smooth planarhelical solutions of Gromeka’s problem in some domain D ? R ³. In the case of domains D with cylindrical boundaries whose generators are orthogonal to R ², it is shown that the choice of a specific solution from the constructed class is reduced to the Dirichlet problem with respect to two functions that are harmonic conjugates in D ² = D∩R ²; i.e., Gromeka’s nonlinear problem is reduced to linear boundary value problems. As an example, a specific solution of the problem for an axisymmetric layer is presented. The solution is based on solving Dirichlet problems in the form of series uniformly convergent in \(\bar D^2\) in terms of wavelet systems that form bases of various spaces of functions harmonic in D ².     

Boundary integral equations for the Helmholtz equation: The third boundary value problem

In this paper we study the application of boundary integral equation methods for the solution of the third, or Robin, boundary value problem for the exterior Helmholtz equation. In contrast to earlier work, the boundary value problem is interpreted here in a weak sense which allows data to be specified in L_∞ (?D), ?D being the boundary of the exterior domain which we assume to be Lyapunov of index 1. For this exterior boundary value problem, we employ Green's theorem to derive a pair of boundary integral equations which have a unique simultaneous solution. We then show that this solution yields a solution of the original exterior boundary value problem.     

On the immersion of digraphs in cubes

We consider some problems concerning generalizations of embeddings of acyclic digraphs inton-dimensional dicubes. In particular, we define an injectioni from a digraphD into then-dimensional dicubeQ _n to be animmersion if for any two elementsa andb inD there is a directed path inD froma tob iff there is a directed path inQ _n fromi(a) toi(b). We further define the immersion to bestrong iff there is a way of choosing these paths so that paths inQ _n corresponding to arcs inD have disjoint interiors, and we introduce a natural notion of “minimality” on the set of arcs of a digraph in terms of its paths. Our main theorem then becomes:Every (minimal) n-element acyclic digraph can be (strongly) immersed in Q _n. We also present examples ofn-element digraphs which cannot be immersed inQ _n?1 and examples of 9n-element non-minimal digraphs which cannot be strongly immersed inQ10n _?1. We conclude with some applications.     

A finite element method scheme for boundary value problems with noncoordinated degeneration of input data

A finite element method scheme is constructed for boundary value problems with noncoordinated degeneration of input data and singularity of a solution. We look at a rate with which an approximate solution by the proposed finite element method converges toward an exact R _ν -generalized solution in the weight set W _2,ν*+β ^2+1/1 (Ω, δ), and establish estimates for the finite element approximation.     

Centerlines of regions in the sphere

Given a region U in the 2-sphere S such that the boundary of U contains at least two points, let D(U) be the collection of open circular disks (called maximal disks) in U whose boundary meets the boundary of U in at least two points and let U₂ be the collection of all regions U⊂S such that for each D∈D(U), D meets the boundary of U in at most two points. In this paper we study geometric properties of regions U∈U₂. We show for such U that the centerline (i.e., the set of centers of maximal disks) is always a smooth, connected 1-manifold. We also show that the boundary of U has at most two components and, if it has exactly two components, then the boundary is locally connected.These results are related the set of points E(X,Y) which are equidistant to two disjoint closed sets X and Y. In particular we investigate when the equidistant set is a 1-manifold.     

Spectral problems in Lipschitz domains

The paper is devoted to spectral problems for strongly elliptic second-order systems in bounded Lipschitz domains. We consider the spectral Dirichlet and Neumann problems and three problems with spectral parameter in conditions at the boundary: the Poincaré–Steklov problem and two transmission problems. In the style of a survey, we discuss the main properties of these problems, both self-adjoint and non-self-adjoint. As a preliminary, we explain several facts of the general theory of the main boundary value problems in Lipschitz domains. The original definitions are variational. The use of the boundary potentials is based on results on the unique solvability of the Dirichlet and Neumann problems. In the main part of the paper, we use the simplest Hilbert L ₂-spaces H _s , but we describe some generalizations to Banach spaces H ^s _p of Bessel potentials and Besov spaces B ^s _p at the end of the paper.     

Estimate of solution stability in a two-dimensional inverse problem for elasticity equations

The problem of determining the density of the medium and one of its elasticity moduli is considered. Properties of the elastic medium and external forces are assumed to be independent of the coordinate x ₃. In this case, the third component of the displacement vector satisfies a scalar equation of the second order, which contains the density ρ of the medium and elasticity modulus μ as coefficients. The parameters ρ and μ are known to be positive and constant everywhere outside some compact domain D ? ?², but they are unknown inside D. The problem of determining these coefficients in D via information, given on the boundary of the domain D for some finite time interval, about a solution of two direct problems is considered. An estimate of the conditional stability of a solution of the inverse problem under consideration is established.     

The Mixed Problem for the Laplacian in Lipschitz Domains

We consider the mixed boundary value problem, or Zaremba’s problem, for the Laplacian in a bounded Lipschitz domain Ω in R ⁿ , n?≥?2. We decompose the boundary $ \partial \Omega= D\cup N$ with D and N disjoint. The boundary between D and N is assumed to be a Lipschitz surface in $\partial \Omega$ . We find an exponent q ₀?>?1 so that for p between 1 and q ₀ we may solve the mixed problem for L ^p . Thus, if we specify Dirichlet data on D in the Sobolev space W ^1,p (D) and Neumann data on N in L ^p (N), the mixed problem with data f _D and f _N has a unique solution and the non-tangential maximal function of the gradient lies in $L^p( \partial \Omega)$ . We also obtain results for p?=?1 when the data comes from Hardy spaces.     

Positive solutions of m-point boundary value problems for a class of nonlinear fractional differential equations

This paper deals with the existence and multiplicity of positive solutions for a class of nonlinear fractional differential equations with m-point boundary value problems. We obtain some existence results of positive solution by using the properties of the Green’s function, u ₀-bounded function and the fixed point index theory under some conditions concerning the first eigenvalue with respect to the relevant linear operator.     

Inverse problems and index formulae for Dirac operators

We consider a Dirac-type operator DP on a vector bundle V over a compact Riemannian manifold (M,g) with a non-empty boundary. The operator DP is specified by a boundary condition P(u_|∂M)=0 where P is a projector which may be a non-local, i.e., a pseudodifferential operator. We assume the existence of a chirality operator which decomposes L²(M,V) into two orthogonal subspaces X₊⊕X₋. Under certain conditions, the operator DP restricted to X₊ and X₋ defines a pair of Fredholm operators which maps X₊→X₋ and X₋→X₊ correspondingly, giving rise to a superstructure on V. In this paper we consider the questions of determining the index of DP and the reconstruction of and DP from the boundary data on ∂M. The data used is either the Cauchy data, i.e., the restrictions to ∂M×R₊ of the solutions to the hyperbolic Dirac equation, or the boundary spectral data, i.e., the set of the eigenvalues and the boundary values of the eigenfunctions of DP. We obtain formulae for the index and prove uniqueness results for the inverse boundary value problems. We apply the obtained results to the classical Dirac-type operator in M×C⁴, M⊂R³.     

Boundary Value Problems for the Elliptic Sine-Gordon Equation in a Semi-strip

We study boundary value problems posed in a semistrip for the elliptic sine-Gordon equation, which is the paradigm of an elliptic integrable PDE in two variables. We use the method introduced by one of the authors, which provides a substantial generalization of the inverse scattering transform and can be used for the analysis of boundary as opposed to initial-value problems. We first express the solution in terms of a 2×2 matrix Riemann–Hilbert problem whose “jump matrix” depends on both the Dirichlet and the Neumann boundary values. For a well posed problem one of these boundary values is an unknown function. This unknown function is characterised in terms of the so-called global relation, but in general this characterisation is nonlinear. We then concentrate on the case that the prescribed boundary conditions are zero along the unbounded sides of a semistrip and constant along the bounded side. This corresponds to a case of the so-called linearisable boundary conditions, however, a major difficulty for this problem is the existence of non-integrable singularities of the function q _y at the two corners of the semistrip; these singularities are generated by the discontinuities of the boundary condition at these corners. Motivated by the recent solution of the analogous problem for the modified Helmholtz equation, we introduce an appropriate regularisation which overcomes this difficulty. Furthermore, by mapping the basic Riemann–Hilbert problem to an equivalent modified Riemann–Hilbert problem, we show that the solution can be expressed in terms of a 2×2 matrix Riemann–Hilbert problem whose “jump matrix” depends explicitly on the width of the semistrip L, on the constant value d of the solution along the bounded side, and on the residues at the given poles of a certain spectral function denoted by h(λ). The determination of the function h remains open.     

The method of boundary integral equations for a mixed scattering problem

In this paper we consider the scattering of an electromagnetic time-harmonic plane wave by an infinite cylinder having an open arc and a bounded domain in R² as cross section. To this end, we solve a scattering problem for the Helmholtz equation in R² where the scattering object is a combination of a crack Γ and a bounded obstacle D, and we have Dirichlet-impedance type boundary condition on Γ and Dirichlet boundary condition on ∂D (∂D∈C²). Applying potential theory, the problem can be reformulated as a boundary integral system. We establish the existence and uniqueness of a solution to the system by using the Fredholm theory.     

Fold completeness of a system of root vectors of a system of unbounded polynomial operator pencils in Banach spaces. II. Application

This paper is a continuation of the author’s paper in 2009,where the abstract theory of fold completeness in Banach spaces has been presented.Using obtained there abstract results,we consider now very general boundary value problems for ODEs and PDEs which polynomially depend on the spectral parameter in both the equation and the boundary conditions.Moreover,equations and boundary conditions may contain abstract operators as well.So,we deal,generally,with integro-differential equations,functional-differential equations,nonlocal boundary conditions,multipoint boundary conditions,integro-differential boundary conditions.We prove n-fold completeness of a system of root functions of considered problems in the corresponding direct sum of Sobolev spaces in the Banach Lq-framework,in contrast to previously known results in the Hilbert L 2-framework.Some concrete mechanical problems are also presented.     

Relaxation and regularization of nonconvex variational problems

We are interested in variational problems of the form min ∝W(?u) dx, withW nonconvex. The theory of relaxation allows one to calculate the minimum value, but it does not determine a well-defined “solution” since minimizing sequences are far from unique. A natural idea for determining a solution is regularization, i.e. the addition of a higher order term such as ε|??u|². But what is the behavior of the regularized solution in the limit as ε→0? Little is known in general. Our recent work [19, 20, 21] discusses a particular problem of this type, namely min _{u y}=±1 ∝∝u _x ² +ε|u _yy|dxdy with various boundary conditions. The present paper gives an expository overview of our methods and results.     

Dual semigroups and second order linear elliptic boundary value problems

It is shown that general second order elliptic boundary value problems on bounded domains generate analytic semigroups onL ₁. The proof is based on Phillips’ theory of dual semigroups. Several sharp estimates for the corresponding semigroups inL _p, 1≦p<∞, are given.