Robin Problem Research Articles

The Robin Problems in the Coupled System of Wave Equations on a Half-Line

This article investigates the local well-posedness of a coupled system of wave equations on a half-line, with a particular emphasis on Robin boundary conditions within Sobolev spaces. We provide estimates for the solutions to linear initial-boundary-value problems related to the coupled system of wave equations, utilizing the Unified Transform Method in conjunction with the Hadamard norm while considering the influence of external forces. Furthermore, we demonstrate that replacing the external force with a nonlinear term alters the iteration map defined by the unified transform solutions, making it a contraction map in a suitable solution space. By employing the contraction mapping theorem, we establish the existence of a unique solution. Finally, we show that the data-to-solution map is locally Lipschitz continuous, thus confirming the local well-posedness of the coupled system of wave equations under consideration.

The Bayesian Approach to Inverse Robin Problems

The Bayesian Approach to Inverse Robin Problems

Well-Posedness of the Schrödinger–Korteweg–de Vries System with Robin Boundary Conditions on the Half-Line

The Schrödinger–Korteweg–de Vries (SKdV) system can describe the nonlinear dynamics of phenomena such as Langmuir and ion acoustic waves, which are highly valuable for studying wave behavior and interactions. The SKdV system has wide-ranging applications in physics and applied mathematics. In this article, we investigate the local well-posedness of the SKdV system with Robin boundary conditions and polynomial terms in the Sobolev space. We want to enhance the applicability of this type of SKdV system. Our verification process is as follows: We estimate Fokas solutions for the Robin problem with external forces. Next, we define an iteration map in suitable solution space and prove the iteration map is a contraction mapping and onto some closed ball B(0,r). Finally, by the contraction mapping theorem, we obtain the uniqueness solution. Moreover, we show that the data-to-solution map is locally Lipschitz continuous and conclude with the well-posedness of the SKdV system.

Publisher Correction to: A class of semilinear hypoelliptic Robin problems and Morse theory

Publisher Correction to: A class of semilinear hypoelliptic Robin problems and Morse theory

Robin problem involving the $p(x)$-Laplacian operator without Ambrosetti-Rabinowizt condition

The paper deals with the following Robin problem$$\left\lbrace\begin{aligned}- \mathcal{M} \left( \int _{\Omega} \frac{1}{p(x)} \vert \nabla u \vert ^{p(x)} dx + \int _{\partial \Omega } \frac{a(x)}{p(x)} \vert \nabla u \vert ^{p(x)} d \sigma \right) \mathop{\rm div} (\vert \nabla u \vert ^{p(x)-2} \nabla u) = \lambda h(x,u) \ \ \text{ in } \Omega,\\\vert \nabla u \vert ^{p(x)-2} \frac{\partial u}{\partial \nu} + a(x) \vert u \vert ^{p(x)-2} u &=0 \quad \quad \quad \ \text{ on } \partial \Omega .\end{aligned}\right.$$The goal is to determine the precise positive interval of $\lambda $ for which the problem admits at least two nontrivial solutions via variational approach for the above problem without assuming the Ambrosetti-Rabinowitz condition. Next, we give a result on the existence of an unbounded sequence of nontrivial weak solutions by employing the fountain theoreom with Cerami condition.

A class of semilinear hypoelliptic Robin problems and Morse theory

A class of semilinear hypoelliptic Robin problems and Morse theory

Nontrivial solutions for a general p(x)-Laplacian Robin problem

We establish the existence of multiple nontrivial solutions for a class of $p(x)$-Laplacian Robin problem. Our approach relies on the variable exponent theory of generalized Lebesgue-Sobolev spaces, combined withadequate variational methods and a variant of the Mountain Pass lemma.

Existence of three solutions to a p(z)-Laplacian-Like Robin problem

Existence of three solutions to a p(z)-Laplacian-Like Robin problem

On the Multiplicity of Solutions of a Discrete Robin Problem with Variable Exponents

On the Multiplicity of Solutions of a Discrete Robin Problem with Variable Exponents

Global Multiplicity of Positive Solutions for Nonlinear Robin Problems with an Indefinite Potential Term

We consider a Robin problem driven by a nonlinear nonhomogeneous differential operator plus an indefinite potential term. The reaction is parametric and exhibits the competing effects of a concave (sublinear) and of a convex (superlinear) terms (“concave-convex” problem). The parameter multiplies the convex term. We prove an existence and multiplicity theorem which is global in parameter.

The Robin problems for the coupled system of reaction–diffusion equations

This article investigates the local well-posedness of Turing-type reaction–diffusion equations with Robin boundary conditions in the Sobolev space. Utilizing the Hadamard norm, we derive estimates for Fokas unified transform solutions for linear initial-boundary value problems subject to external forces. Subsequently, we demonstrate that the iteration map, defined by the unified transform solutions and incorporating nonlinearity instead of external forces, acts as a contraction map within an appropriate solution space. Our conclusive result is established through the application of the contraction mapping theorem.

On a nonlinear Robin problem with an absorption term on the boundary and L 1 data

Abstract We deal with existence and uniqueness of nonnegative solutions to: − Δ u = f ( x ) , in Ω , ∂ u ∂ ν + λ ( x ) u = g ( x ) u η , on ∂ Ω , \left\{\begin{array}{ll}-\Delta u=f\left(x),\hspace{1.0em}& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega ,\\ \frac{\partial u}{\partial \nu }+\lambda \left(x)u=\frac{g\left(x)}{{u}^{\eta }},\hspace{1.0em}& \hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.33em}\partial \Omega ,\end{array}\right. where η ≥ 0 \eta \ge 0 and f , λ f,\lambda , and g g are the nonnegative integrable functions. The set Ω ⊂ R N ( N > 2 ) \Omega \subset {{\mathbb{R}}}^{N}\left(N\gt 2) is open and bounded with smooth boundary, and ν \nu denotes its unit outward normal vector. More generally, we handle equations driven by monotone operators of p p -Laplacian type jointly with nonlinear boundary conditions. We prove the existence of an entropy solution and check that, under natural assumptions, this solution is unique. Among other features, we study the regularizing effect given to the solution by both the absorption and the nonlinear boundary term.

On Pleijel's nodal domain theorem for the Robin problem

AbstractWe prove an improved Pleijel nodal domain theorem for the Robin eigenvalue problem. In particular, we remove the restriction, imposed in previous work, that the Robin parameter be non‐negative. We also improve the upper bound in the statement of the Pleijel theorem. In the particular example of a Euclidean ball, we calculate the explicit value of the Pleijel constant for a generic constant Robin parameter, and we show that it is equal to the Pleijel constant for the Dirichlet Laplacian on a Euclidean ball.

Nonlinear Robin problems with double phase variable exponent operator

Nonlinear Robin problems with double phase variable exponent operator

Robin problems for elliptic equations with singular drifts on Lipschitz domains

Robin problems for elliptic equations with singular drifts on Lipschitz domains

Sound radiation of a vibrating circular plate set in a hemispherical enclosure

This study rigorously addresses the Neumann–Robin problem of sound radiation from a vibrating flexible disk embedded into a hemispherical enclosure. The enclosure is acoustically soft on its inside surface and hard on its outside surface. The method of superimposed solutions, expressed in terms of Dini series and spherical harmonics, has been applied for this purpose. The presented solution is applicable to any vibration velocity profile on the radiator, with an elastically supported thin plate excited asymmetrically used as an example. The results presented here can be valuable for modeling vibroacoustic responses of various speakers, sensors, and microphones with edges either clamped to the enclosure or nearly free. Achieving this involves modifying the values of the boundary stiffness for resisting transverse displacement and the rotation of the plate’s edge. Additionally, when applying these results to model the responses of alarm or signaling transducers, high-amplitude responses are expected for selected resonant frequencies. In such cases, the boundary stiffness values can be set to ensure multiple strong and wide resonant maxima in the frequency characteristics. In addition, the results presented in this study include solving the dynamic equation of motion for the plate or rigid piston with external excitation. This equation is coupled with the wave equation inside and outside the hemispherical enclosure, allowing the inclusion of the enclosure’s effect on the plate-enclosure system’s responses. This approach can be useful for modeling and improving the acoustic responses of spherical speakers. In such cases, the boundary stiffness values should be relatively small to emulate a soft suspension of the plate. The impedance of the enclosure’s internal surface can be arbitrarily selected, ranging from acoustically hard to acoustically soft. In this study, the impedance of mineral wool has been applied. Numerical analysis is presented for frequencies below 4kHz, with errors smaller than 0.2%. The results have been validated using the finite element method.

On inhomogeneous exterior Robin problems with critical nonlinearities

The paper studies the large-time behavior of solutions to the Robin problem for PDEs with critical nonlinearities. For the considered problems, nonexistence results are obtained, which complements the interesting recent results by Ikeda et al. (2020) [7], where critical cases were left open. Moreover, our results provide partially answers to some other open questions previously posed by Zhang (2001) [21] and Jleli and Samet (2019) [13].

On approximation of the Dirichlet problem for divergence form operators by Robin problems

We show that, under natural assumptions, solutions of Dirichlet problems for uniformly elliptic divergence form operators can be approximated pointwise by solutions of some versions of Robin problems. The proof is based on the stochastic representation of solutions and properties of reflected diffusions corresponding to divergence form operators.

Asymptotic analysis of perturbed Robin problems in a planar domain

We consider a perforated domain \(\Omega(\epsilon)\) of \(\mathbb{R}^2\) with a small hole of size \(\epsilon\) and we study the behavior of the solution of a mixed Neumann-Robin problem in \(\Omega(\epsilon)\) as the size \(\epsilon\) of the small hole tends to \(0\). In addition to the geometric degeneracy of the problem, the nonlinear \(\epsilon\)-dependent Robin condition may degenerate into a Neumann condition for \(\epsilon=0\) and the Robin datum may diverge to infinity. Our goal is to analyze the asymptotic behavior of the solutions to the problem as \(\epsilon\) tends to \(0\) and to understand how the boundary condition affects the behavior of the solutions when \(\epsilon\) is close to \(0\). The present paper extends to the planar case the results of [36] dealing with the case of dimension \(n\geq 3\).
 For more information see https://ejde.math.txstate.edu/Volumes/2023/57/abstr.html

Boundary value problems on non-Lipschitz uniform domains: stability, compactness and the existence of optimal shapes

We study boundary value problems for bounded uniform domains in R n , n ⩾ 2, with non-Lipschitz, and possibly fractal, boundaries. We prove Poincaré inequalities with uniform constants and trace terms for ( ε , ∞ )-domains contained in a fixed bounded Lipschitz domain. We introduce generalized Dirichlet, Neumann, and Robin problems for Poisson-type equations and prove the Mosco convergence of the associated energy functionals along sequences of suitably converging domains. This implies a stability result for weak solutions, the norm convergence of the associated resolvents, and the convergence of the corresponding eigenvalues and eigenfunctions. We provide compactness results for parametrized classes of admissible domains, energy functionals, and weak solutions. Using these results, we can then prove the existence of optimal shapes in these classes in the sense that they minimize the initially given energy functionals. For the Robin boundary problems, this result is new.