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The authors consider the existence, multiplicity, and uniqueness for polyharmonic problem with Navier boundary conditions. One of the interesting features in our proof is that we give a new attempt to consider the uniqueness of nontrivial solution for the above polyharmonic problem by using the theory of monotone mappings. This is probably the first time this theory is used to solve polyharmonic problems. Then we apply the fixed point theorems on cones to analyze the existence and multiplicity of positive solutions for the above polyharmonic problem. This is very difficult for partial differential equations, especially for polyharmonic equations. The main reason is that the Green's function for the above polyharmonic problem is unbounded. We overcome the difficulties by using some new techniques. The uniqueness of nontrivial solution and the existence of positive solutions for polyharmonic equations with Dirichlet boundary conditions are also considered. [ABSTRACT FROM AUTHOR]

Highlights • Large PHS+poly based RBF-FD stencils can lead to high orders of accuracy without numerical ill-conditioning. • It can also combine high orders of accuracy near boundaries with an absence of Runge-phenomenon-type boundary errors. • Numerical explanations to this behavior are provided based on a closed-form expression for the RBF+poly cardinal functions. • It explains the role of polynomials and RBFs in RBF+poly approximations. Abstract Radial basis function generated finite difference (RBF-FD) approximations generalize grid-based regular finite differences to scattered node sets. These become particularly effective when they are based on polyharmonic splines (PHS) augmented with multi-variate polynomials (PHS+poly). One key feature is that high orders of accuracy can be achieved without having to choose an optimal shape parameter and without having to deal with issues related to numerical ill-conditioning. The strengths of this approach were previously shown to be especially striking for approximations near domain boundaries, where the stencils become highly one-sided. Due to the polynomial Runge phenomenon, regular FD approximations of high accuracy will in such cases have very large weights well into the domain. The inclusion of PHS-type RBFs in the process of generating weights makes it possible to avoid this adverse effect. With that as motivation, this study aims at gaining a better understanding of the behavior of PHS+poly generated RBF-FD approximations near boundaries, illustrating it in 1-D, 2-D and 3-D. [ABSTRACT FROM AUTHOR]

We study the unique (non-unique) solvability of the Neumann problem for the polyharmonic equation in unbounded domains under the assumption that a generalized solution of this problem has a bounded Dirichlet integral with weight . Depending on the value of the parameter a, we prove uniqueness (or non-uniqueness) theorems or present exact formulas for the dimension of the solution space of the Neumann problem in the exterior of a compact set and in a half space. [ABSTRACT FROM AUTHOR]

We consider the Laplacian in a planar infinite straight strip with frequent alternation of boundary conditions. We show that for a sufficiently small alternation period, there are no gaps in a lower part of the spectrum. In terms of certain numbers and functions, we write an explicit upper bound for the period and an expression for the length of the lower part of the spectrum in which the absence of gaps is guaranteed. [ABSTRACT FROM AUTHOR]

In the present paper we establish the Wiener test for boundary regularity of the solutions to the polyharmonic operator. We introduce a new notion of polyharmonic capacity and demonstrate necessary and sufficient conditions on the capacity of the domain responsible for the regularity of a polyharmonic function near a boundary point. In the case of the Laplacian the test for regularity of a boundary point is the celebrated Wiener criterion of 1924. It was extended to the biharmonic case in dimension three by Mayboroda and Maz'ya (Invent Math 175(2):287-334, 2009). As a preliminary stage of this work, in Mayboroda and Maz'ya (Invent Math 196(1):168, 2014) we demonstrated boundedness of the appropriate derivatives of solutions to the polyharmonic problem in arbitrary domains, accompanied by sharp estimates on the Green function. The present work pioneers a new version of capacity and establishes the Wiener test in the full generality of the polyharmonic equation of arbitrary order. [ABSTRACT FROM AUTHOR]

In this paper, we analyse the polyharmonic equationin a bounded smooth domain with homogeneous Dirichlet conditions on the boundary and critical exponents. We will obtain a relationship between the topological domain and the multiplicity of solutions. We also give symmetry properties of the solutions and a result about breaking symmetry. [ABSTRACT FROM AUTHOR]

We extend holomorphically polyharmonic functions on a real ball to a complex set being the union of rotated balls. We solve a Dirichlet type problem for complex polyharmonic functions with the boundary condition given on the union of rotated spheres. [ABSTRACT FROM AUTHOR]

We study the properties of fractional integro-differential operators. As an application, we analyze the solvability of some boundary value problems for the inhomogeneous polyharmonic equation in the unit ball. These problems generalize the Dirichlet and Neumann problems to the case of fractional boundary operators. [ABSTRACT FROM AUTHOR]

We consider the inverse boundary value problem for the first order perturbation of the polyharmonic operator L g , X , q , with X being a W 1 , ∞ vector field and q being an L ∞ function on compact Riemannian manifolds with boundary which are conformally embedded in a product of the Euclidean line and a simple manifold. We show that the knowledge of the Dirichlet-to-Neumann map determines X and q uniquely. The method is based on the construction of complex geometrical optics solutions using the Carleman estimate for the Laplace–Beltrami operator due to Dos Santos Ferreira, Kenig, Salo and Uhlmann. Notice that the corresponding uniqueness result does not hold for the first order perturbation of the Laplace–Beltrami operator. [ABSTRACT FROM AUTHOR]

In this paper, we give three kinds of Almansi representations for super-polyharmonic functions. As applications of these representations, we investigate two kinds of boundary value problems in superspace and construct solutions to the super-Poisson equation and the inhomogeneous super-polyharmonic equation. [ABSTRACT FROM PUBLISHER]

This work is devoted to the study of the boundary value problem ( − 1 ) α Δ α u = ( − 1 ) k S k [ u ] + λ f , x ∈ Ω ⊂ R N , u = ∂ n u = ∂ n 2 u = ⋯ = ∂ n α − 1 u = 0 , x ∈ ∂ Ω , where the k -Hessian S k [ u ] is the k th elementary symmetric polynomial of eigenvalues of the Hessian matrix and the datum f obeys suitable summability properties. We prove the existence of at least two solutions, of which at least one is isolated, strictly by means of variational methods. We look for the optimal values of α ∈ N that allow the construction of such an existence and multiplicity theory and also investigate how a weaker definition of the nonlinearity permits improving these results. [ABSTRACT FROM AUTHOR]

We find a polynomial solution of the Dirichlet problem with polynomial boundary data for the polyharmonic equation with polynomial right-hand side in the unit ball. [ABSTRACT FROM AUTHOR]

This work is devoted to the solvability of some non classical boundary value problems for the polyharmonic equation. On the boundary we consider operators of fractional order in the Riemann–Liouville and Hadamard sense. The considered problems generalize the Dirichlet and the Neumann problems with boundary operators of a fractional order. [ABSTRACT FROM AUTHOR]

In this article, we consider the following quasilinear polyharmonic equation: Δ n/ m m u = λ h( x)| u| q-1 u + u| u| p e| u|β in Ω, u = ∇ u = ⋯ = ∇ m-1 u = 0 on ∂Ω, where Ω ⊂ ℝ n, n ≥ 2 m ≥ 2, is a bounded domain with smooth boundary. The real-valued function h is a sign-changing and unbounded function. The exponents p, q and β satisfy 0 < q < n/( m-1) < p+1, β ∈ (1, n/( n- m)] and λ > 0. Using the Nehari manifold and fibering maps, we show the existence and multiplicity of solutions. [ABSTRACT FROM AUTHOR]

We obtain solvability conditions for the Neumann problem for the homogeneous polyharmonic equation in the unit ball. We study the arithmetic triangle that arises in these conditions. For the elements of that triangle, we obtain recursion relations similar to the relations for the Pascal, Euler, and Stirling triangles. We establish a relationship between the Neumann problem and the Dirichlet problem. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following weighted PDE system with Navier boundary conditions on a half space:1Whenis any even number betweenand, we establish the equivalence between (1) and the following integral system2under some very mild growth condition, whereis the reflection of the pointabout the plane. Then, in the critical case of, we show that every pair of positive solutionsof system (2) is rotationally symmetric about-axis. While in the subcritical case, we prove the non-existence of positive solutions. When dealing integral system (2),can be any real number betweenand. [ABSTRACT FROM PUBLISHER]

We prove a dichotomy result giving the positivity preserving property for a biharmonic equation with Dirichlet boundary conditions arising in MEMS models. We adapt some ideas in [H.-Ch. Grunau, G. Sweers, Positivity for equations involving polyharmonic operators with Dirichlet boundary conditions, Math. Ann. 307 (1997), 589-626]. [ABSTRACT FROM AUTHOR]

We study the arithmetic triangle arising from the solvability conditions for the Neumann problem for the polyharmonic equation in the unit ball. Recurrence relations for the elements of this triangle are obtained. [ABSTRACT FROM AUTHOR]

We show that a first order perturbation A(x) · D + q(x) of the polyharmonic operator (-?)m, m = 2, can be determined uniquely from the set of the Cauchy data for the perturbed polyharmonic operator on a bounded domain in Rn, n = 3. Notice that the corresponding result does not hold in general when m = 1. [ABSTRACT FROM AUTHOR]

We derive and explicitly solve recurrent equations with boundary conditions for coefficients in the solvability condition for the Neumann problem for the polyharmonic equation in a unit ball. Bibliography: 9 titles. [ABSTRACT FROM AUTHOR]

A coretraction, a right inverse to the trace operator to a smooth hypersurface 𝒮 with the smooth boundary Γ = ∂𝒮 ≠ ∅, is constructed. The coretraction extends m-tuples of functions from the Besov spaces on each face of 𝒮 into the Bessel potential space on the domain slit by the hypersurface , provided these tuples satisfy a compatibility conditions on the boundary Γ. The traces are defined by arbitrary Dirichlet systems of boundary operators and extension is performed by two different methods, one implicit and one explicit. Explicit extension, based on the solution to the Dirichlet BVP for the poly-harmonic equation, permits the extension of distributions from the Besov space with a negative s < 0. Moreover, it permits to establish some additional features of the extended functions, which are useful in applications. Coretractions have essential applications in boundary value problems for partial differential equations when, for example, it is necessary to reduce a BVP with non-homogeneous boundary conditions to a BVP with the homogeneous boundary conditions. [ABSTRACT FROM AUTHOR]

In a bounded connected domain, we obtain boundary conditions for the volume potential for the polyharmonic equation. [ABSTRACT FROM AUTHOR]

Abstract: Let be a positive integer. We investigate the existence and the asymptotic behavior of positive continuous solutions to the following semilinear polyharmonic boundary value problem in the unit ball of : where and is a positive measurable function in such that there exists satisfying for each , , and is a continuous function on with . [Copyright &y& Elsevier]

Abstract: This paper proposes a meshless method based on coupling the method of fundamental solutions (MFS) with quasi-interpolation for the solution of nonhomogeneous polyharmonic problems. The original problems are transformed to homogeneous problems by subtracting a particular solution of the governing differential equation. The particular solution is approximated by quasi-interpolation and the corresponding homogeneous problem is solved using the MFS. By applying quasi-interpolation, problems connected with interpolation can be avoided. The error analysis and convergence study of this meshless method are given for solving the boundary value problems of nonhomogeneous harmonic and biharmonic equations. Numerical examples are also presented to show the efficiency of the method. [Copyright &y& Elsevier]

We prove existence results for positive bounded continuous solutions of a nonlinear polyharmonic system by using a potential theory approach and properties of a large functional class Km,n called Kato class. [ABSTRACT FROM AUTHOR]

Abstract: We determine boundary remainder terms for some higher order Hardy–Rellich inequalities involving the polyharmonic operator . The results are proved by studying suitable auxiliary boundary eigenvalue problems, the optimal constants found may not be the classical Hardy–Rellich ones. [Copyright &y& Elsevier]

We aim at giving a rich class of quasi-metrics from which we obtain as an application an interesting inequality for the Greens function of the fractional Laplacian in a smooth domain in Rn. [ABSTRACT FROM AUTHOR]

The representation of solutions in an explicit form for boundary value problems of differential equations is always very important. In this article the Green function for the Dirichlet problem is constructed in an explicit form inside a sphere for the polyharmonic equations. We note that the obtained formulas have an independent sense. In particular, the explicit representation of the solution of the Dirichlet problem for the polyharmonic equation has important consequences in the theory of elasticity. [ABSTRACT FROM AUTHOR]

In this article, the Dirichlet problem for polyharmonic functions investigated in (Begehr, H., Wang, Y. and Du, J., A Dirichlet problem for polyharmonic functions (In press)) is also solved by a new approach different from the reflection method used in (Begehr, H., Wang, Y. and Du, J., A Dirichlet problem for polyharmonic functions (In press)). The explicit expression of the unique solution for the Dirichlet problem of triharmonic functions is obtained by using the so-called weak decomposition of polyharmonic functions and turning the problem into an equivalent Dirichlet boundary value problem for polyanalytic functions. In contrast to the boundary condition according to Begehr, Wang and Du, the requirement of smoothness for the given functions is reduced. [ABSTRACT FROM AUTHOR]

Here we study the polyharmonic nonlinear elliptic boundary value problem on the unit ball B in ℝn (n ≥ 2) (- Δ)mu + g(·,u) = 0, in B (in the sense of distributions) limx→Ɛϵ∂B(u(x)/(1 - ∣x∣²)m-1) = 0(Ɛ). Under appropriate conditions related to a Kato class on the nonlinearity g(x, t), we give some existence results. Our approach is based on estimates for the polyharmonic Green function on B with zero Dirichlet boundary conditions, including a 3G-theorem, which leeds to some useful properties on functions belonging to the Kato class. [ABSTRACT FROM AUTHOR]

We establish a criterion for existence of an L2-limit and a weak L2-limit of a polyharmonic function on a regular analytic boundary of a bounded plane domain. [ABSTRACT FROM AUTHOR]

In this paper we give the results obtained by the constructed probabilistic representation of the solution to the first boundary value problem for a polyharmonic equation. It is shown that this solution is expressed by the parametric derivative of the solution to the specially constructed Dirichlet problem for the Helmholtz equation. On this basis we develop new algorithms of 'walk on a grid' and 'random walk by spheres' for the solution of a polyharmonic equation. [ABSTRACT FROM AUTHOR]

We formulate and solve the 3-d analog of the second basic problem for a cylindrical elastic solid. [Copyright &y& Elsevier]

We study polyharmonic boundary value problems (−Δ)mu=f(u), m∈N, with Dirichlet boundary conditions on bounded and unbounded conformally contractible domains in Rn. Such domains can be contracted to a point (bounded case) or to infinity (unbounded case) by one-parameter groups of conformal maps. The class of star-shaped domain is a subclass. The problem has variational structure. This allows us to derive a sufficient condition for uniqueness by studying the interaction of one-parameter transformation groups with the underlying functional L. If the transformation group strictly reduces the values of L then uniqueness of the critical point of L follows. The proof is inspired by E. Noether''s theorem on symmetries and conservation laws. Applications of the uniqueness principle are given in Part II of this paper. [Copyright &y& Elsevier]

We present existence results for the polyharmonic nonlinear elliptic boundary-value problem (- Δ)mu = f (&mdot;, u) in B (∂/∂&nu)ju = 0 on ∂B, 0 ≤ j ≤ m - 1. (in the sense of distributions, where B is the unit ball in Rn and n ≥ 2. The nonlinearity f(x, t) satisfies appropriate conditions related to a Kato class of functions Km,n. Our approach is based on estimates for the polyharmonic Green function with zero Dirichlet boundary conditions and on the Schauder fixed point theorem. [ABSTRACT FROM AUTHOR]

By means of a perturbation method recently introduced by Bolle, we discuss the existence of infinitely many solutions for a class of perturbed symmetric higher order Schrödinger equations with non-homogeneous boundary data on unbounded domains. [ABSTRACT FROM AUTHOR]