Robin Problem Research Articles

Temperature on rods with Robin boundary conditions

We consider solutions uf to the one-dimensional Robin problem with the heat source f∈L1[−π,π] and Robin parameter α>0. For given m, M, and s, 0≤m<s<M, we identify the heat sources f0, such that uf0 maximizes the temperature gap max[−π,π]⁡uf−min[−π,π]⁡uf over all heat sources f such that m≤f≤M and ‖f‖L1=2πs. In particular, this answers a question raised by J. J. Langford and P. McDonald in [5]. We also identify heat sources, which maximize/minimize uf at a given point x0∈[−π,π] over the same class of heat sources as above and discuss a few related questions.

Positive solutions for a class of nonlinear parametric Robin problems

We consider a nonlinear Robin problem driven by the p-Laplacian and a parametric concave-convex reaction with the parameter multiplying the convex (superlinear) term. We prove a multiplicity result for positive solutions which is global in the parameter λ>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda >0$$\\end{document} (bifurcation-type theorem). We also show the existence of a minimal positive solution uλ∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$u_{\\lambda }^*$$\\end{document} and determine the monotonicity and continuity properties of the map λ⟼uλ∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda \\longmapsto u_{\\lambda }^*$$\\end{document}

High-order iterative scheme to the Robin problem for a nonlinear wave equation with viscoelastic term

The report deals with the Robin problem for a nonlinear wave equation with viscoelastic term. Under some suitable conditions, we establish a high-order iterative scheme and then prove that the scheme converges to the weak solution of the original problem along with the error estimate. This result extends the result in [9].

Boundary value problems of potential theory for the exterior ball and the approximation and ergodic behaviour of the solutions

The paper is concerned with the interconnection of the boundary behaviour of the solutions of the exterior Dirichlet, Neumann and Robin problems of harmonic analysis for the unit ball in R3 with the corresponding behaviour of the associated ergodic inverse problems for the entire space. Rates of approximation play a basic role.The solutions themselves are evaluated by means of Fourier expansions with respect to spherical harmonics. In case of the first two problems, the basis for the investigation of the approximation and ergodic behaviour is the theory of semigroups of linear operators mapping a Banach space X into itself. The connection between the semigroup property and the major premise of Huygens’ principle is emphasized.Another tool is a Drazin-like inverse operator Aad for the infinitesimal generator A of a semigroup that arises naturally in ergodic theory. This operator is a closed, not necessarily bounded, operator. It was introduced in a paper with U. Westphal (Butzer and Westphal, 1970/71) and extended to a generalized setting with J.J. Koliha (Butzer and Koliha, 2009).Unlike the latter two problems, the solution of Robin’s problem does not have the semigroup property and therefore the semigroup methods applied to Dirichlet’s and Neumann’s problem do not work. The authors give several hints how to overcome these difficulties.

Multiplicity Results for Nonlinear Nonhomogeneous Robin Problems with Indefinite Potential Term

Multiplicity Results for Nonlinear Nonhomogeneous Robin Problems with Indefinite Potential Term

Weighted symmetrization results for a problem with variable Robin parameter

By means of a suitable weighted rearrangement, we obtain various apriori bounds for the solutions to a Robin problem. Among other things, we derive a family of Faber-Krahn type inequalities.

Infinitely many smooth nodal solutions for Orlicz Robin problems

In this note, we study a Robin problem driven by the Orlicz g-Laplace operator. In particular, by using a regularity result and Kajikiya’s theorem, we prove that the problem has a whole sequence of distinct smooth nodal solutions converging to the trivial one. The analysis is developed in the most general abstract setting that corresponds to Orlicz–Sobolev function spaces.

Boundedness of solutions to Dirichlet, Neumann and Robin problems for elliptic equations in Orlicz spaces

Boundary value problems for second-order elliptic equations in divergence form, whose nonlinearity is governed by a convex function of non-necessarily power type, are considered. The global boundedness of their solutions is established under boundary conditions of Dirichlet, or Neumann, or Robin type. A decisive role in the results is played by optimal forms of Orlicz-Sobolev embeddings and boundary trace embeddings, which allow for critical growths of the coefficients.

COMPLEX BOUNDARY VALUE PROBLEMS FOR THE CAUCHY–RIEMANN OPERATOR ON A TRIANGLE

There are three over-determined boundary value problems for the inhomogeneous Cauchy–Riemann equation, the Dirichlet problem, the Neumann problem and the Robin problem. These problems have found their significant importance and applications in diverse fields of science, for example, applied mathematics, physics, engineering and medicine. In this paper, we explicitly investigate the solvability of these three basic boundary value problems for the Cauchy–Riemann operator on an isosceles orthogonal triangle. The solvability conditions are explicitly obtained. The plane parqueting reflection principle is used. Cauchy–Pompeiu-type formulae for the triangle are established. The boundary behavior of a related integral operator of the Schwarz type is studied in detail. Then, the boundary values of solutions at the corner points are found.

Multiple ground-state solutions with sign information for double-phase robin problems

Multiple ground-state solutions with sign information for double-phase robin problems

Existence of Weak Solutions for Weighted Robin Problem Involving $$p\left( .\right)$$-biharmonic operator

Existence of Weak Solutions for Weighted Robin Problem Involving $$p\left( .\right)$$-biharmonic operator

Weak solution to a Robin problem of anomalous diffusion equations: Uniqueness and stable algorithm for the TPC system

A Riemann‐Liouville fractional Robin boundary‐value problem is proposed to describe the fast heat transfer law both within isotropic materials and through the boundary of the materials in high temperature environment. The variational formulation of the fractional model is derived, and further, the energy estimation of the weak solution is deduced. Subsequently, the uniqueness theorem of weak solution is proved. A valid and stable finite difference scheme is developed for the fractional model. The numerical experiments are implemented to indicate that the fractional model is applicable to discover the thermal superdiffusion in the thermal protective clothing (TPC) system and numerical algorithms are effective to improve the intelligence of TPC design.

On a generalization of the robin problem for the Laplace equation in the circle

In this paper, we study the solvability of the fractional analogue of the Robin problem for the Laplace equation. A modified fractional differentiation operator in the sense of Hadamard is considered as a boundary operator. Boundary conditions are given in the form of a connection between different values of the unknown function in a circle. The problem is solved using the Fourier expansion method. For various values of the parameters of the boundary operators involved, theorems on the existence and uniqueness of a solution to the problem under study are proved.

Interaction of scales for a singularly perturbed degenerating nonlinear Robinproblem.

We study the asymptotic behaviour of solutions of a boundary value problem for the Laplace equation in a perforated domain in , , with a (nonlinear) Robin boundary condition on the boundary of the small hole. The problem we wish to consider degenerates in three respects: in the limit case, the Robin boundary condition may degenerate into a Neumann boundary condition, the Robin datum may tend to infinity, and the size ϵ of the small hole where we consider the Robin condition collapses to 0. We study how these three singularities interact and affect the asymptotic behaviour as ϵ tends to 0, and we represent the solution and its energy integral in terms of real analytic maps and known functions of the singular perturbation parameters.This article is part of the theme issue ‘Non-smooth variational problems and applications’.

The Robin and Neumann problems for the nonlinear Schrödinger equation on the half-plane

This work studies the initial-boundary value problem (ibvp) of the two-dimensional nonlinear Schrödinger equation on the half-plane with initial data in Sobolev spaces and Neumann or Robin boundary data in appropriate Bourgain spaces. It establishes well-posedness in the sense of Hadamard by using the explicit solution formula for the forced linear ibvp obtained via Fokas’s unified transform, and a contraction mapping argument.

On the Fučík spectrum of the [formula omitted]-Laplacian with no-flux boundary condition

In this paper, we study the quasilinear elliptic problem −Δpu=au+p−1−bu−p−1inΩ,u=constanton∂Ω,0=∫∂Ω|∇u|p−2∇u⋅νdσ,where the operator is the p-Laplacian and the boundary condition is of type no-flux. In particular, we consider the Fučík spectrum of the p-Laplacian with no-flux boundary condition which is defined as the set Πp of all pairs (a,b)∈R2 such that the problem above has a nontrivial solution. It turns out that this spectrum has a first nontrivial curve C being Lipschitz continuous, decreasing and with a certain asymptotic behavior. Since (λ2,λ2) lies on this curve C, with λ2 being the second eigenvalue of the corresponding no-flux eigenvalue problem for the p-Laplacian, we get a variational characterization of λ2. This paper extends corresponding works for Dirichlet, Neumann, Steklov and Robin problems.

Arbitrarily small nodal solutions for nonhomogeneous Robin problems

In the present paper, we consider a nonlinear Robin problem driven by a nonhomogeneous differential operator and with a reaction which is only locally defined. Using cut-off techniques and variational tools, we show that the problem has a sequence of nodal solutions converging to zero in C 1 ( Ω ‾ ).

Fractional double phase Robin problem involving variable‐order exponents and logarithm‐type nonlinearity

The paper deals with the logarithmic fractional equations with variable exponents where and denote the variable ‐order ‐fractional Laplace operator and the nonlocal normal ‐derivative of ‐order, respectively, with and ( ) being continuous. Here, is a bounded smooth domain with ( ) for any and are a positive parameters, and are two continuous functions, while variable exponent can be close to the critical exponent , given with and for . Precisely, we consider two cases. In the first case, we deal with subcritical nonlinearity, that is, , for any . In the second case, we study the critical exponent, namely, for some . Then, using variational methods, we prove the existence and multiplicity of solutions and existence of ground state solutions to the above problem.

Nonexistence of positive solutions for nonlinear parabolic Robin problems and Hardy–Leray inequalities

Nonexistence of positive solutions for nonlinear parabolic Robin problems and Hardy–Leray inequalities

Fractional double phase Robin problem involving variable order-exponents without Ambrosetti–Rabinowitz condition

We consider a fractional double phase Robin problem involving variable order and variable exponents. The nonlinearity f is a Carathéodory function satisfying some hypotheses which do not include the Ambrosetti–Rabinowitz-type condition. By using a variational methods, we investigate the multiplicity of solutions.