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The purpose of this work is to study the following elliptic problem: $$\begin{aligned} (\mathrm{P}_\lambda )\qquad \left\{ \begin{array}{ll} -\Delta u =f(x)|u(x)|^{p-2}u(x) +\lambda g(x)|u|^{q-2}u \;\; \text{ in } \,\Omega ;\\ u = 0 \;\; \text{ in } \,\partial \Omega , \end{array} \right. \end{aligned}$$ where $$\Omega \subset {\mathbb {R}}^N\;(N\ge 3)$$ be a bounded smooth domain, $$f,\,g\in L^\infty (\Omega ),$$ $$\lambda $$ is a positive parameter. Under adequate assumptions on the sources terms f and g, we establish the existence of three solutions: one is positive, one is negative and the other changes sign.

In this paper we investigate the problem of recovering the source term in an elliptic system from a measurement of the state on a part of the boundary. For the particular interest in reconstructing probably discontinuous sources, we use the standard least squares method with the total variation regularization. The finite element method is then applied to discretize the minimization problem, we show the stability and the convergence of this technique. Furthermore, we have proposed an algorithm to stably solve the minimization problem. We prove the iterate sequence generated by the derived algorithm converging to a minimizer of the regularization problem, and that convergence measurement is also established. Finally, a numerical experiment is presented to illustrate our theoretical findings., Inverse source problem, boundary observation, total variation regularization, ill-posedness, finite element method, stability and convergence, elliptic boundary value problem

An existence theory is developed for an elliptic boundary value problem in image analysis known as edge-enhancing diffusion (EED) inpainting. The EED inpainting problem aims at restoration of missing data in an image as the steady state of a nonlinear anisotropic diffusion process where the known data provide Dirichlet boundary conditions. The existence of a weak solution is established by applying the Leray–Schauder fixed point theorem, and it is shown that the set of all possible weak solutions is bounded. Moreover, it is demonstrated that under certain conditions the sequences resulting from iterative application of the operator from the existence theory contain convergent subsequences.

We study a nonlinear elliptic boundary value problem defined on a smooth bounded domain involving the fractional Laplace operator, a concave-convex powers term together with mixed Dirichlet-Neumann boundary conditions.

We propose a high-order adaptive numerical solver for the semilinear elliptic boundary value problem modeling magnetic plasma equilibrium in axisymmetric confinement devices. In the fixed boundary case, the equation is posed on curved domains with piecewise smooth curved boundaries that may present corners. The solution method we present is based on the hybridizable discontinuous Galerkin method and sidesteps the need for geometry-conforming triangulations thanks to a transfer technique that allows to approximate the solution using only a polygonal subset as computational domain. Moreover, the solver features automatic mesh refinement driven by a residual-based a posteriori error estimator. As the mesh is locally refined, the computational domain is automatically updated in order to always maintain the distance between the actual boundary and the computational boundary of the order of the local mesh diameter. Numerical evidence is presented of the suitability of the estimator as an approximate error measure for physically relevant equilibria with pressure pedestals, internal transport barriers, and current holes on realistic geometries.

Using the variational approach and the critical point theory, we established several criteria for the existence of at least one nontrivial solution for a discrete elliptic boundary value problem with a weight $p(\cdot, \cdot)$ and depending on a real parameter $\lambda$.

One way to generalize the boundary Yamabe problem posed by Escobar is to ask if a given metric on a compact manifold with boundary can be conformally deformed to have vanishing σ k -curvature in the interior and constant H k -curvature on the boundary. When restricting to the closure of the positive k-cone, this is a fully nonlinear degenerate elliptic boundary value problem with fully nonlinear Robin-type boundary condition. We prove nonuniqueness for the boundary-value problem σ 4 -curvature equals zero and constant H 4 -curvature by using bifurcation results proven by Case, Moreira and Wang. Surprisingly, our construction via products of sphere and hyperbolic space only works for a finite set of dimensions.

We present a 3×3 Riemann-Hilbert problem formalism for the initial-boundary value problem of the two-component nonlinear Schrodinger (2-NLS) equation on the half-line. And we also get the Dirichlet to Neuemann map through analysising the global relation in this paper.

We propose an immersed hybrid difference method for elliptic boundary value problems by artificial interface conditions. The artificial interface condition is derived by imposing the given boundary condition weakly with the penalty parameter as in the Nitsche trick and it maintains ellipticity. Then, the derived interface problems can be solved by the hybrid difference approach together with a proper virtual to real transformation. Therefore, the boundary value problems can be solved on a fixed mesh independently of geometric shapes of boundaries. Numerical tests on several types of boundary interfaces are presented to demonstrate efficiency of the suggested method.

We study the elliptic sinh-Gordon equation formulated in the quarter plane by using the so-called Fokas method, which is a significant extension of the inverse scattering transform for the boundary value problems. The method is based on the simultaneous spectral analysis for both parts of the Lax pair and the global algebraic relation that involves all boundary values. In this paper, we address the existence theorem for the elliptic sinh-Gordon equation posed in the quarter plane under the assumption that the boundary values satisfy the global relation. We also present the formal representation of the solution in terms of the unique solution of the matrix Riemann- Hilbert problem defined by the spectral functions.

For any pair of smooth vector-valued functions of 3d field theory we consider an unconditional formula connecting the values of the normal derivatives, curls, and divergence on the boundary of a domain and show that the corresponding boundary value problem is well posed.

In this paper, we study a three point boundary value problem of nonlinear fractional differential equations of order α, 2 < α < 3. New existence and uniqueness results are established using Banach fixed point theorem. Other existence results are obtained using Schaefer and Krasnoselskii theorems. Some illustrative examples are also presented.

The Neumann problem for a second-order parabolic equation with integro-differential operator in the boundary condition is considered. A well-posedness theorem is proved, in particular, the integral representation of the solution is obtained, estimates for the derivatives of the solution are established, and the kernel of the inverse operator of the problem is explicitly expressed.

We study the uniqueness of the solution of a boundary value problem for the biharmonic equation in unbounded domains under the assumption that the generalized solution of this problem has a bounded Dirichlet integral with weight |x| a . Depending on the value of the parameter a, we prove uniqueness theorems or present exact formulas for the dimension of the solution space of this problem in the exterior of a compact set and in a half-space.

We study a boundary value problem for an equation of mixed type with the Lavrent’ev–Bitsadze operator in the leading part and with variable deviation of the argument in lower-order terms. The general solution of the equation is constructed. We prove a uniqueness theorem without any conditions on the value of the deviation. The problem is uniquely solvable. We derive integral representations of the solutions in closed form in the elliptic and hyperbolic domains.

We consider a boundary value transmission problem for two-dimensional filtration flows in an anisotropic porous layer consisting of adjacent domains in which the media have essentially different conductivities (permeability and thickness). In general, the layer conductivity is specified by a nonsymmetric second rank tensor whose components are modeled by continuously differentiable functions of coordinates. To study the problem, we use two complex planes, the physical plane and an auxiliary plane, which are related by a homeomorphic (one-to-one and continuous) transformation satisfying an equation of the Beltrami type. On the physical plane, we pose a transmission problem for a rather complicated elliptic system of equations. This problem is reduced on the auxiliary plane to canonical form, which dramatically simplifies the analysis of the problem. Then the problem is reduced to a system of boundary singular integral equations with generalized kernels of the Cauchy type, which are expressed via the fundamental solutions of the main equations. The boundary value transmission problem studied here can be used as a mathematical model of processes arising in the recovery of fluids (water and oil) from natural soil formations of complicated geological structure.

The regularity of solutions to the Dirichlet boundary value problem is studied for elliptic differential operators of order 2m defined on subdomains of a manifold. Relationships between the smoothness of the right hand side, the boundary, and the solutions are obtained.

We consider a three-dimensional boundary value problem for the Laplace equation on a thin plane screen with boundary conditions for the “directional derivative”: boundary conditions for the derivative of the unknown function in the directions of vector fields defined on the screen surface are posed on each side of the screen. We study the case in which the direction of these vector fields is close to the direction of the normal to the screen surface. This problem can be reduced to a system of two boundary integral equations with singular and hypersingular integrals treated in the sense of the Hadamard finite value. The resulting integral equations are characterized by the presence of integral-free terms that contain the surface gradient of one of the unknown functions. We prove the unique solvability of this system of integral equations and the existence of a solution of the considered boundary value problem and its uniqueness under certain assumptions.

The Dirichlet problem in the half-plane for elliptic equations containing, in addition to differential operators, the shift operator with respect to the variable parallel to the boundary of the domain is considered. The behavior of the solution as the variable orthogonal to the boundary of the domain increases unboundedly is studied.

This paper is concerned with the existence of solutions to the following singular elliptic boundary value problem involving \({p}\)-Laplace operator $$-{\rm div}(|\nabla u|^{p-2} \nabla u)=\frac{h}{u^\gamma} \text{in } \Omega, \quad u > 0\text{ in } \Omega,\quad u=0 \text{ on } \partial\Omega.$$ Here, \({\Omega\subset \mathbb{R}^N(N\geq3)}\) is a bounded domain with smooth boundary, and \({h}\) is a positive \({L^1}\) function on \({\Omega}\). A “compatibility condition” on the couple \({(h(x),\gamma)}\) is given for the problem to have at least one solution. More precisely, it is shown that the problem admits at least one solution if and only if there exists a \({u_0\in W_0^{1,p}(\Omega)}\) such that \({\int_\Omega hu_0^{1-\gamma} \mathrm{d}x < \infty}\). This generalizes a previous result obtained by Sun and Zhang (Calc Var Partial Differ Equ 49:909–922, 2014) who considered the case \({p=2}\).

For a singularly perturbed elliptic boundary value problem, an asymptotic expansion of the boundary-layer solution is constructed and justified in the case when the boundary layer consists of three zones with different behavior of the solution, which is caused by the multiplicity of the root of the degenerate equation.

In this paper we study the multiplicity of nontrivial solutions for a superlinear elliptic boundary value problem. By combining truncation techniques, minimax theorem, Morse theory and bifurcation method, we show that the problem has at least seven nontrivial solutions and give sign information for them. No symmetry is assumed on the domain or the nonlinearity.

This paper is mainly related to multiple nontrivial solutions of the elliptic boundary value problem { − Δ u = | u | p − 2 u + f ( x , u ) , x ∈ Ω , u = 0 , x ∈ ∂ Ω , for p ∈ ( 2 , 2 ⁎ ) . It is reasonable to guess that for dim ⁡ Ω ≥ 2 above problem possesses infinitely many distinct solutions since this is proved to be true for ODE. However, so far one does not even know if there exists a fourth nontrivial solution. By using a new homological linking theorem, Morse theory, and some precise estimates we disclose the relationship among the gaps of consecutive eigenvalues of Laplace operator, growth trend of nonlinear terms and the existence of multiple solutions of superlinear elliptic boundary value problem. Moreover, as p is close to 2, we get the fourth nontrivial solution under appropriate hypotheses, where f ( x , u ) satisfies Ambrosetti–Rabinowitz condition.

In this paper, the Dirichlet-type problem for the system of elliptic equations of second order with the degeneracy at a line crossing the domain is considered. The Dirichlet-type problem with additionally given asymptotics of the solution at this line is discussed. The uniqueness and the existence of the solution of this problem in the class of Holder functions is proved.

We consider second order uniformly elliptic operators of divergence form in \(\mathbb {R}^{d+1}\) whose coefficients are independent of one variable. For such a class of operators we establish a factorization into a product of first order operators related with Poisson operators and Dirichlet–Neumann maps. Consequently, we obtain a solution formula for the inhomogeneous elliptic boundary value problem in the half space, which is useful to show the existence of solutions in a wider class of inhomogeneous data. We also establish \(L^2\) solvability of boundary value problems for a new class of the elliptic operators.

In this paper we examine a numerical implementation of Fokas' unified method for elliptic boundary value problems on convex polygons. Within this setting the unified method provides a reconstruction of the unknown boundary values from the known boundary data for an important class of elliptic boundary value problems. We demonstrate that this approach can be used to study the classical Dirichlet eigenvalue problem for the Laplacian on convex polygons. We show that this problem is equivalent to a nonlinear eigenvalue problem for a semi-Fredholm operator which has holomorphic dependence on the eigenvalue itself. We also study the classical Dirichlet problem for the Helmholtz operator. We provide a new Galerkin scheme for the underlying Dirichlet-Neumann map, and prove that for sufficiently regular Dirichlet data this scheme converges with spectral accuracy.

Boundary value problems are considered on a simplex F in the real Euclidean space R2. The recent discovery of new families of special functions, orthogonal on F, makes it possible to consider not only the Dirichlet or Neumann boundary value problems on F, but also the mixed boundary value problem which is a mixture of Dirichlet and Neumann type, ie. on some parts of the boundary of F a Dirichlet condition is fulfilled and on the other Neumann’s works.

We consider the problem { div u + 〈 a ; u 〉 = f in Ω u = u 0 on ∂ Ω . We show that if curl a ( x 0 ) ≠ 0 for some x 0 ∈ Ω , then the problem is solvable without restriction on f . We also discuss the regularity of the solution.

We obtain a new upper estimate for the Green’s function associated with a higher order fractional boundary value problem. As an application of this result, criteria for the existence of positive solutions of the problem are then established.

We study the Neumann problem in a bounded domain Ω for a scalar elliptic equation with coefficients oscillating with respect to two groups of variables with periods of different smallness order. We prove the H1- and L2-estimates of homogenization. The estimates have the operator form and can be expressed in terms of the resolvent of the original problem and its approximations in the operator (L2(Ω)→H1(Ω))- and (L2(Ω)→L2(Ω))- norms.

We consider the Dirichlet problem for Laplace’s equation with real-analytic data posed on the boundary of a cylinder. If the base of the cylinder is an ellipsoid and if the data function is an entire function of order less than 1, we show that there is an entire solution.

In the paper, a Lagrange optimal control problem governed by a fractional Dirichlet problem with the Riemann---Liouville derivative is considered. To begin with, based on some variational method, the existence and continuous dependence of solution to the aforementioned Dirichlet problem is investigated. Then, continuous dependence is applied to show the existence of optimal solution to the Lagrange problem. An important point is that the solution to Dirichlet problem does need to be unique; therefore, the above dependence should be understood as a continuity of some multifunction--the concept of the Kuratowski---Painleve limit of the sequence of sets is used to formulate this property.

In the field of shape and topology optimization the new concept is the topological derivative of a given shape functional. The asymptotic analysis is applied in order to determine the topological derivative of shape functionals for elliptic problems. The topological derivative (TD) is a tool to measure the influence on the specific shape functional of insertion of small defect into a geometrical domain for the elliptic boundary value problem (BVP) under considerations. The domain with the small defect stands for perturbed domain by topological variations. This means that given the topological derivative, we have in hand the first order approximation with respect to the small parameter which governs the volume of the defect for the shape functional evaluated in the perturbed domain. TD is a function defined in the original (unperturbed) domain which can be evaluated from the knowledge of solutions to BVP in such a domain. This means that we can evaluate TD by solving only the BVP in the intact domain. One can consider the first and the second order topological derivatives as well, which furnish the approximation of the shape functional with better precision compared to the first order TD expansion in perturbed domain. In this work the topological derivative is applied in the context of Electrical Impedance Tomography (EIT). In particular, we are interested in reconstructing a number of anomalies embedded within a medium subject to a set of current fluxes, from measurements of the corresponding electrical potentials on its boundary. The basic idea consists in minimize a functional measuring the misfit between the boundary measurements and the electrical potentials obtained from the model with respect to a set of ball-shaped anomalies. The first and second order topological derivatives are used, leading to a non-iterative second order reconstruction algorithm. Finally, a numerical experiment is presented, showing that the resulting reconstruction algorithm is very robust with respect to noisy data.

We propose certain interpretation of boundary conditions on , the boundary of , when f belongs to the weighted Sobolev space subordinated to the integrable weight and g is defined on , where is a given domain. It may happen that the trace embedding theorems are missing. Our proposed interpretation requires to have defined an extension operator , where X is the relevant function space of functions defined on . We show that the proposed interpretation does not depend on the choice of the extension operator . Moreover, we apply our result to obtain existence and uniqueness of solutions to an example of the boundary value problem involving degenerated p-Laplacian with nonhomogeneous boundary condition. Existence of the extension operator within the class of weights when is certain weighted Slobodetskii space and is a Lipschitz boundary domain is confirmed by our previous results.

We define an R?-generalized solution of the first boundary value problem for a second-order elliptic equation with degeneration of the input data on the entire boundary of the two-dimensional domain and prove the existence and uniqueness of the solution in the weighted set Open image in new window.

The aim of this paper is to obtain the existence of solution for the fractional p-Laplacian Dirichlet problem with mixed derivatives t D T α(|0 D t α u(t)| p-2 0 D t α u(t)) = f(t,u(t)), t ∈ [0,T], u(0) = u(T) = 0, where 1/p < α < 1, 1 < p < ∞ and f : [0,T] × ℝ → ℝ is a Carathéodory function which satisfies some growth conditions. We obtain the existence of nontrivial solutions by using the direct method in variational methods and mountain pass theorem.

In the paper we study questions about solvability of some boundary value problems for a non-homogenous poly-harmonic equation. As a boundary operator we consider differentiation operator of fractional order in Miller-Ross sense. The considered problem is a generalization of well-known Dirichlet and Neumann problems.

Some nonstandard boundary value problems are studied for the stationary Poisson system, Stokes system, and Navier–Stokes system. The problems under consideration are “intermediate” between the Dirichlet problem and Neumann problem. The well-posedness of these problems is proved.

We elaborate a boundary Fourier method for studying an analogue of the Hilbert problem for analytic functions within the framework of generalised Cauchy-Riemann equations. The boundary value problem need not satisfy the Shapiro-Lopatinskij condition and so it fails to be Fredholm in Sobolev spaces. We show a solvability condition of the Hilbert problem, which looks like those for ill-posed problems, and construct an explicit formula for approximate solutions.

A theorem of existence and non-existence of solutions for a boundary value problem for the equations of axially symmetric gravitational field in vacuum is given using the inverse function theorem in Banach spaces and the method of functional solutions. Conditions are given under which solutions exist or not exist.

A boundary-value problem of finding eigenvalues is considered for the negative Laplace operator in a disk with Neumann boundary condition on almost all the circle except for a small arc of vanishing length, where the Dirichlet boundary condition is imposed. A complete asymptotic expansion with respect to a parameter (the length of the small arc) is constructed for an eigenvalue of this problem that converges to a double eigenvalue of the Neumann problem.

A Dirichlet nonlinear problem for a second-order equation is considered on an interval. The problem is perturbed by a delta-like potential e −1 Q(e −1 x), where the function Q(ξ) is compactly supported and 0 < e ≪ 1. A solution of this boundary value problem is constructed with accuracy up to O(e) with the use of the method of matched asymptotic expansions. The obtained asymptotic approximation is validated by means of the fixed-point theorem. All types of boundary conditions are considered for a linear boundary value problem.

Several approaches are discussed how to understand the solution of the Dirichlet problem for the Poisson equation when the Dirichlet data are non-smooth such as if they are in L 2 only. For the method of transposition (sometimes called very weak formulation) three spaces for the test functions are considered, and a regularity result is proved. An approach of Berggren is recovered as the method of transposition with the second variant of test functions. A further concept is the regularization of the boundary data combined with the weak solution of the regularized problem. The effect of the regularization error is studied. The regularization approach is the simplest to discretize. The discretization error is estimated for a sequence of quasi-uniform meshes. Since this approach turns out to be equivalent to Berggren's discretization his error estimates are rendered more precisely. Numerical tests show that the error estimates are sharp, in particular that the order becomes arbitrarily small when the maximal interior angle of the domain tends to 2 π .© 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1433–1454, 2016