Adjoint and self-adjoint boundary value problems on a geometric graph

We consider boundary value problems of arbitrary order for linear differential equations on a geometric graph. Solutions of boundary value problems are coordinated at the interior vertices of the graph and satisfy given conditions at the boundary vertices. For considered boundary value problems, we construct adjoint boundary value problems and obtain a self-adjointness criterion. We describe the structure of the solution set of homogeneous self-adjoint boundary value problems with alternating coefficients of a differential equation and obtain nondegeneracy conditions for these boundary value problems.     

On output functionals of boundary value problems on stochastic domains

We consider the computation of output functionals of random solutions to elliptic boundary value problems in domains with random boundary perturbations. We use a second‐order shape calculus to linearize the problem around a fixed nominal domain. For known mean and two‐point correlation function of the boundary perturbation, we derive, with leading order, deterministic expressions for the mean and the variance of the random output functional. These expressions include the solution of the boundary value problem on the nominal domain and a further, deterministic solution of the so‐called adjoint equation. The theoretical findings are supported and quantified by numerical experiments. Copyright © 2009 John Wiley & Sons, Ltd.     

Generalized solutions and generalized eigenfunctions of boundary-value problems on a geometric graph

We consider generalized solutions to boundary-value problems for elliptic equations on an arbitrary geometric graph and their corresponding eigenfunctions. We construct analogs of Sobolev spaces that are dense in L ₂. We obtain conditions for the Fredholm solvability of boundary-value problems of various types, describe their spectral properties and conditions for the expansion in generalized eigenfunctions. The results presented here are fundamental in studying boundary control problems of oscillations of multiplex jointed structures consisting of strings or rods, as well as in studying the cell metabolism.     

Existence and Ulam's type stability results for a class of fractional boundary value problems on a star graph

This paper is concerned with a nonlinear fractional boundary value problem on a star graph. By using a transformation, the suggested problem is converted into an equivalent system of fractional boundary value problem. Schaefer's fixed point theorem and Banach's contraction principle is used to establish its existence and uniqueness results. Further, different kinds of Ulam's type stability results for the proposed problem have been discussed. Finally, two examples are presented to illustrate the application of the obtained results.     

Contractibility of level sets of functionals associated with some elliptic boundary value problems and applications

Assume that I is the functional defined on the Hilbert space concerning the problem: in and u on , where f is sublinear at and superlinear at 0, that is, , and is the first eigenvalue of in . Under very general conditions, I has at least two local minimizers u ₁ and u ₂ and one mountain pass point u ₃ , and . Assuming that u ₁ , u ₂ and u ₃ are the only three nontrivial critical points of I, we prove that the level set I _b is contractible for all . Using this conclusion, we extend one of Hofer's result concerning existence of four nontrivial solutions of the above problem to the case where I is not and the trivial critical point 0 may be degenerate. Since I is not , the local topological degree and the critical groups of u ₃ can not be clearly computed. The lack of topological information about 0 and u ₃ makes it impossible to use topological degree theory or Morse theory in obtaining the fourth nontrivial solution. To overcome these difficulties, we explore a new technique in this paper.     

Maximum principles for functionals associated with the solution of semilinear elliptic boundary value problems

We consider a new P-function associated with the solutionu of an elliptic boundary value problem and obtain pointwise bounds for the gradient in terms of the maximum ofu and the geometry of the domain. SimilarP-functions have previously been used to obtain bounds of the same type. Our results give improved bounds for certain problems, in particular we obtain isoperimetric inequalities for the maximum stress in the Saint-Venant torsion problem.     

Firefighting on a random geometric graph

LetG_λ be the graph whose vertices are points of a planar Poisson process of density λ, with vertices adjacent if they are within distance 1. A “fire” begins at some vertex and spreads to all neighbors in discrete steps; in the meantime f vertices can be deleted at each time‐step. Let f_λ be the least f such that, with probability 1, any fire on G_λ can be stopped in finite time. We show that f_λ is bounded between two linear functions of λ. The lower bound makes use of a new result concerning oriented percolation in the plane; the constant factor in the upper bound is is tight, provided a certain conjecture, for which we offer supporting evidence, is correct. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 46, 466–477, 2015     

Quadratic functionals of Brownian motion

Functionals of Brownian motion can be dealt with by realizing them as functionals of white noise. Specifically, for quadratic functionals of Brownian motion, such a realization is a powerful tool to investigate them. There is a one-to-one correspondence between a quadratic functional of white noise and a symmetric L²(R²)-function which is considered as an integral kernel. By using well-known results on the integral operator we can study probabilistic properties of quadratic or certain exponential functionals of white noise. Two examples will illustrate their significance.     

Nonlinear boundary value problems

In this paper we consider two quasilinear boundary value problems. The first is vector valued and has periodic boundary conditions. The second is scalar valued with nonlinear boundary conditions determined by multivalued maximal monotone maps. Using the theory of maximal monotone operators for reflexive Banach spaces and the Leray-Schauder principle we establish the existence of solutions for both problems.     

On elastostatic boundary value problems with a conical boundary point

We study elastostatic boundary value problems with a conical boundary point by the method of integral equations. The equations of such problems are singular. In the case of a smooth surface, we construct a regularizer for these equations; in the case of a surface with a conical point, the regularizer is constructed in such a way as to ensure that the kernel of the regularized equation belongs to the class B and satisfies the assumptions of the Fredholm alternative theorem. We analyze the properties of elastic potentials in the case of a surface with a conical point.     

Singularly perturbed boundary value problems with a power-law boundary layer

We consider boundary value problems for second-order singularly perturbed equations whose solution has a power-law boundary layer that occurs because the degenerate equation has multiple roots.     

Quadratic variation functionals and dilation equations

Here we present some distribution function inequalities between certain functionals defined relative to a convolution approximation procedure. Such inequalities are best known when the approximation is made using dilations of the Gaussian or Cauchy kernels. In these cases, classical differential equations, the heat equation or Laplace's equation, provide the basis for comparisons; in the latter case, the quadratic functional is known as the Lusin area integral. The kernels we consider are compactly supported, and satisfy a dilation equation, rather than a differential equation. For these kernels, there is an intrinsic quadratic variation, defined from the dilation structure. We obtain good lambda distribution function inequalities between a maximal function and the quadratic variation functional.     

On the solvability of a boundary value problem for a fourth-order equation on a graph

We consider a boundary value problem for a fourth-order equation on a graph modeling elastic deformations of a plane rod system with conditions of rigid connection at the vertices. Conditions for the unique solvability are stated. We also present sufficient conditions for the problem to be degenerate.