On the boundary value problem for the spectrally loaded heat conduction operator

We consider the boundary value problems in a quarter-plane for a loaded heat conduction operator (one-dimensional in the space variable). A peculiarity of the operator in question is as follows: first, the spectral parameter is the coefficient of the loaded summand; second, the order of the derivative in the loaded summand is equal to that of the differential part of the operator, and third, the load point moves with a variable velocity. We demonstrate that the boundary value problem under study is Noetherian.     

Approximation of the bifurcation function for elliptic boundary value problems

The bifurcation function for an elliptic boundary value problem is a vector field B(ω) on R ^d whose zeros are in a one‐to‐one correspondence with the solutions of the boundary value problem. Finite element approximations of the boundary value problem are shown to give rise to an approximate bifurcation function B^h(ω), which is also a vector field on R ^d. Estimates of the difference B(ω) − B^h(ω) are derived, and methods for computing B^h(ω) are discussed. © 2000 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 16: 194–213, 2000     

The hilbert boundary value problem for elliptic differential equation in the plane

The solution w to the Hilbert boundary value problem ?w/?z = F(z, w,?w/?z) in D Re(a+ib) w = g on ?D has so far been solved in the space of Holder-continuously differentiable functions C₁α(D). It is shown here that theproblem has a unique solution in the more general Sobolev space W_1,p (D), 2 < p < ∞, provided that the boundaryfunction g is allowed to belong to the Slobodecky space W_s,p (?D), S = 1 ? 1/P.     

Multiple solutions of a quasilinear elliptic problem involving nonlinear boundary condition on exterior domain

In this paper, we study the multiplicity of non‐negative solutions for the quasilinear p‐Laplacian equation with the nonlinear boundary condition (1) where Δ_p denotes the p‐Laplacian operator, defined by △ _pu = div( | ? u | ^{p ? 2} ? u),1 < p < N, Ω is a smooth exterior domain in . is the outward normal derivative, . The parameters p,q,r are either or . The weight functions a(x),h(x),g(x) satisfy some suitable conditions. Using the decomposition of the Nehari manifold and the variational methods, we prove that problem (1) has at least two positive solutions provided 0 < | λ | < λ₁ for some λ₁. Copyright © 2012 John Wiley & Sons, Ltd.     

On solvability of a boundary value problem for a nonhomogeneous biharmonic equation with a boundary operator of a fractional order

This paper is concerned with the solvability of a boundary value problem for a nonhomogeneous biharmonic equation. The boundary data is determined by a differential operator of fractional order in the Riemann-Liouville sense. The considered problem is a generalization of the known Dirichlet and Neumann problems.     

Robin boundary value problem for the Cauchy-Riemann operator

The aim of this article is to give explicit representations for solutions of the Robin boundary value problem for the Cauchy-Riemann operator [image omitted]. In the homogeneous cases we investigate the Robin boundary condition in a more general form. Finally, we give solutions of the corresponding higher-order operators.     

The existence of a generalized fourier transform for a solution as a radiation condition for a class of problems generalizing oscillation excitation problems in regular waveguides

For a second-order inhomogeneous differential equation defined on the real axis and such that its right-hand side and solutions are functions in a Hilbert space, it is shown that the existence of a generalized Fourier transform of the solution is a correct radiation condition if the right-hand side is sufficiently smooth and compactly supported.     

Multiple solutions for a semilinear boundary value problem: a computational multiplicity proof

We prove the first genuine “partial differential equation” result on a conjecture concerning the number of solutions of second-order elliptic boundary value problems with a nonlinearity which grows superlinearly at +∞. The proof makes massive use of computer assistance: After approximate solutions have been computed by a numerical mountain pass algorithm, combined with a Newton iteration to improve accuracy, a fixed point argument is used to show the existence of exact solutions close to the approximations.     

具p-Laplace算子的分数阶微分方程边值问题的正解

考虑具有p-Laplace算子的分数阶泛函微分方程边值问题,利用锥上的不动点定理,得到了其正解及多个正解存在的充分条件,所得结果推广了已有的结论,并举例说明了结论的适用性.     

Full discretization of a nonlinear parabolic problem containing volterra operators and an unknown dirichlet boundary condition

The reconstruction of an unknown solely time‐dependent Dirichlet boundary condition in a nonlinear parabolic problem containing a linear and a nonlinear Volterra operator is considered. The inverse problem is converted into a variational problem in which the unknown Dirichlet condition is eliminated using a given integral overdetermination. A time‐discrete recurrent approximation scheme is designed, using Backward Euler's method. The convergence of the approximations towards a solution of the variational problem is proved under appropriate assumptions on the data and on the Volterra operators. The uniqueness of this solution is shown in the case that the nonlinear Volterra operator satisfies a particular inequality. Moreover, the Finite Element Method is used to discretize the time‐discrete approximation scheme in space. Finally, full‐discrete error estimates are derived for a particular choice of the finite elements. The corresponding convergence rates are supported by a numerical experiment. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1444–1460, 2015     

A nonlinear problem for the Laplace equation with a degenerating Robin condition

We investigate the behavior of the solutions of a mixed problem for the Laplace equation in a domain Ω. On a part of the boundary ∂Ω, we consider a Neumann condition, whereas in another part, we consider a nonlinear Robin condition, which depends on a positive parameter δ in such a way that for δ = 0 it degenerates into a Neumann condition. For δ small and positive, we prove that the boundary value problem has a solution u(δ,·). We describe what happens to u(δ,·) as δ→0 by means of representation formulas in terms of real analytic maps. Then, we confine ourselves to the linear case, and we compute explicitly the power series expansion of the solution.     

Asymptotics of eigenvalues of the Dirichlet boundary value problem for the Lame operator in a three-dimensional domain with a small cavity

A boundary value problem for the Lame operator in a bounded three-dimensional domain with a small cavity is studied. The domain is filled with an elastic homogeneous isotropic medium that is clamped at the boundary, which corresponds to the Dirichlet boundary condition. The leading term of an asymptotic expansion for the eigenvalue is constructed in the case of the Dirichlet limit problem. The asymptotic expansion is constructed in powers of a small parameter ? that is the diameter of the cavity.     

The structure of the boundary element matrix for the three-dimensional Dirichlet problem in elasticity

The algebraic properties of the matrix arising for the three-dimensional Dirichlet problem for Lamé equations in a rotational domain by the boundary element method are considered. The use of the special basis leads to a matrix having a block structure with sparse blocks. The possible strategies for the efficient solution of the above problem are discussed. © 1998 John Wiley & Sons, Ltd.     

A basic inequality for the Stokes operator related to the Navier boundary condition

We show that ‖Au+Δu_‖L²(Ωε)?C(ε‖∇u_‖L²(Ωε)+‖u_‖L²(Ωε)), where Ωε is a thin domain in R³ of depth ε, the vector field u belongs to the domain of A, which is the Stokes operator for divergence-free vector fields on Ωε satisfying the Navier boundary condition.     

kth Order convergence for a semilinear elliptic boundary value problem in the divergence form

The convergence problem of approximate solutions for a semilinear elliptic boundary value problem in the divergence form is studied. By employing the method of quasilinearization, a sequence of approximate solutions converging with the kth (k ? 2) order convergence to a weak solution for a semilinear elliptic problem is obtained via the variational approach.     

Reconstruction of inclusions for the inverse boundary value problem with mixed type boundary condition

We consider an inverse boundary value problem for identifying the inclusion inside a known anisotropic conductive medium. We give a reconstruction procedure for identifying the inclusion from the Dirichlet–Neumann map or the Neumann–Dirichlet map associated with the mixed type boundary condition.     

ASYMPTOTIC BEHAVIOR FOR A CLASS OF ELLIPTIC EQUIVALUED SURFACE BOUNDARY VALUE PROBLEM WITH DISCONTINUOUS INTERFACE CONDITIONS

Spontaneous potential well-logging is one of the important techniques in petroleum exploitation. A spontaneous potential satisfies an elliptic equivalued surface boundary value problem with diseontinuous interface conditlons. In practice, the measuring electrode is so small that we can simplify the corresponding equivalued surface to a point. In this paper, we give a positive answer to this approximation process: when the equivalued surface shrinks to a point, the solution of the original equivalued surface boundary value problem converges to the solution of the corresponding limit boundary value problem.