By applying Ricceri's variational principle, we demonstrate the existence of solutions for the following Robin problem
\begin{equation*}\left\{ \begin{array}{cc}-\func{div}\left( ω_{1}(x)\left\vert \nabla u\right\vert^{p(x)-2}\nabla u\right) =λω_{2}(x)f(x,u), & x\in Ω\\ ω_{1}(x)\left\vert \nabla u\right\vert ^{p(x)-2}\frac{\partial u}{ \partial \upsilon }+β(x)\left\vert u\right\vert ^{p(x)-2}u=0, & x\in \partial Ω, \end{array} \right. \end{equation*}
in $W_{ω_{1},ω_{2}}^{1,p(.)}\left( Ω\right) $ under some appropriate conditions.
We study some properties of Laplacian eigenvalues with negative Robin boundary conditions. We will show some monotonicity properties on annuli of the first eigenvalue by means of shape optimization techniques.
In this work, we shall study the nonlinear inverse problems of recovering the Robin coefficients in elliptic and parabolic systems of second order, and establish their local Lipschitz stabilities. Some local Lipschitz stability was derived for an elliptic inverse Robin problem. We shall first restructure the arguments in \cite{chou04} for the local Lipschitz stability so that the stability follows from three basic conditions for the elliptic inverse Robin problem. The new arguments are then generalized to help establish a novel local Lipschitz stability for parabolic inverse Robin problems.
Let $n\ge2$ and $Ω$ be a bounded Lipschitz domain in $\mathbb{R}^n$. In this article, the authors investigate global (weighted) estimates for the gradient of solutions to Robin boundary value problems of second order elliptic equations of divergence form with real-valued, bounded, measurable coefficients in $Ω$. More precisely, let $p\in(n/(n-1),\infty)$. Using a real-variable argument, the authors obtain two necessary and sufficient conditions for $W^{1,p}$ estimates of solutions to Robin boundary value problems, respectively, in terms of a weak reverse Hölder inequality with exponent $p$ or …
In this paper we study some properties of the torsion function with Robin boundary conditions. Here we write the shape derivative of the $L^{\infty}$ and $L^p$ norms, for $p\ge 1$, of the torsion function, seen as a functional on a bounded simply connected open set $Ω\subset \mathbb{R}^n$, and prove that the balls are critical shapes for these functionals, when the volume of $Ω$ is preserved.
We consider a non-autonomous Cauchy problem involving linear operators associated with time-dependent forms $a(t;.,.):V\times V\to {\mathbb{C}}$ where $V$ and $H$ are Hilbert spaces such that $V$ is continuously embedded in $H$. We prove $H$-maximal regularity under a new regularity condition on the form $a$ with respect to time; namely H{ö}lder continuity with values in an interpolation space. This result is best suited to treat Robin boundary conditions. The maximal regularity allows one to use fixed point arguments to some non linear parabolic problems with Robin boundary conditions.
By applying Mountain Pass Lemma, Ekeland's and Ricceri's variational principle, Fountain Theorem, we prove the existence and multiplicity of solutions for the following Robin problem \begin{equation*} \left\{ \begin{array}{cc} -\text{div}\left( a(x)\left\vert \nabla u\right\vert ^{p(x)-2}\nabla u\right) =λb(x)\left\vert u\right\vert ^{q(x)-2}u, & x\in Ω\\ a(x)\left\vert \nabla u\right\vert ^{p(x)-2}\frac{\partial u}{\partial \upsilon }+β(x)\left\vert u\right\vert ^{p(x)-2}u=0, & x\in \partial Ω, \end{array} \right. \end{equation*} under some appropriate conditions in the space $W_{a,b}^{1,p(.)…
We study a Robin boundary problem for degenerate parabolic equation. We suggest a notion of entropy solution and propose a result of existence and uniqueness. Numerical simulations illustrate some aspects of solution behaviour. Monodimensional experiments are presented.
The aim of this paper is twofold: First, we characterize an essentially optimal class of boundary operators $Θ$ which give rise to self-adjoint Laplacians $-Δ_{Θ, Ω}$ in $L^2(Ω; d^n x)$ with (nonlocal and local) Robin-type boundary conditions on bounded Lipschitz domains $Ω\subset\bbR^n$, $n\in\bbN$, $n\geq 2$. Second, we extend Friedlander's inequalities between Neumann and Dirichlet Laplacian eigenvalues to those between nonlocal Robin and Dirichlet Laplacian eigenvalues associated with bounded Lipschitz domains $Ω$, following an approach introduced by Filonov for this type of problems.
In this paper we study the $p$-Poisson equation with Robin boundary conditions, where the Robin parameter is a function. By means of some weighted isoperimetric inequalities, we provide various sharp bounds for the solutions to the problems under consideration. We also derive a Faber-Krahn-type inequality.
We are interested in the inverse problem of recovering a Robin coefficient defined on some non accessible part of the boundary from available data on another part of the boundary in the nonstationary Stokes system. We prove a Lipschitz stability estimate under the \textit{a priori} assumption that the Robin coefficient lives in some compact and convex subset of a finite dimensional vectorial subspace of the set of continuous functions. To do so, we use a theorem proved by L. Bourgeois which establishes Lipschitz stability estimates for a class of inverse problems in an abstract framework.
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 Variational methods, we investigate the multiplicity of solutions.
We consider the problem of minimising the $k$th eigenvalue, $k \geq 2$, of the ($p$-)Laplacian with Robin boundary conditions with respect to all domains in $\mathbb{R}^N$ of given volume $M$. When $k=2$, we prove that the second eigenvalue of the $p$-Laplacian is minimised by the domain consisting of the disjoint union of two balls of equal volume, and that this is the unique domain with this property. For $p=2$ and $k \geq 3$, we prove that in many cases a minimiser cannot be independent of the value of the constant $α$ in the boundary condition, or equivalently of the volume $M$. We obtain …
Comparison results of Talenti type for Elliptic Problems with Dirichlet boundary conditions have been widely investigated in the last decades. In this paper, we deal with Robin boundary conditions. Surprisingly, contrary to the Dirichlet case, Robin boundary conditions make the comparison sensitive to the dimension, and while the planar case seems to be completely settled, in higher dimensions some open problems are yet unsolved.
The paper is devoted to the study of some properties of the first eigenvalue of the anisotropic $p$-Laplace operator with Robin boundary condition involving a function $β$ which in general is not constant. In particular we obtain sharp lower bounds in terms of the measure of the domain and we prove a monotonicity property of the eigenvalue with respect the set inclusion.
In this paper, we study a class of boundary value problems (BVPs) with Robin conditions in some $L^p$ spaces for polyharmonic equation on Lipschitz domains. Utilizing polyharmonic fundamental solutions, these Robin BVPs are solved by the method of layer potentials. The crucial ingedients of our approach are the classical single layer potential and its higher order analog (which are called multi-layer $S$-potentials), and the main results generalize ones of second order (Laplacian) case to higher order (polyharmonic) case.
(Version 3) We study the problem of finding functions, defined within and on an ellipse, whose Laplacian is -1 and which satisfy a homogeneous Robin boundary condition on the ellipse. The parameter in the Robin condition is denoted by beta. The general solution and various asymptotic approximations are obtained. The integral of the solution over the ellipse, denoted by Q, is a quantity of interest in some physical applications. The dependence of Q on beta and the ellipse geometry is found. Several methods are used.
In this paper, we develop some properties of the $a_{x,y}(.)$-Neumann derivative for the fractional $a_{x,y}(.)$-Laplacian operator. Therefore we prove the basic proprieties of the correspondent function spaces. In the second part of this paper, by means of Ekeland's variational principal and direct variational approach, we prove the existence of weak solutions for a nonlocal problem with nonhomogeneous Neumann and Robin boundary condition.
In this paper, we consider the Stokes equations and we are concerned with the inverse problem of identifying a Robin coefficient on some non accessible part of the boundary from available data on the other part of the boundary. We first study the identifiability of the Robin coefficient and then we establish a stability estimate of logarithm type thanks to a Carleman inequality due to A. L. Bukhgeim and under the assumption that the velocity of a given reference solution stays far from 0 on a part of the boundary where Robin conditions are prescribed.
We classify positive solutions to a class of quasilinear equations with Neumann or Robin boundary conditions in convex domains. Our main tool is an integral formula involving the trace of some relevant quantities for the problem. Under a suitable condition on the nonlinearity, a relevant consequence of our results is that we can extend to weak solutions a celebrated result obtained for stable solutions by Casten and Holland and by Matano.
In this paper, we investigate an initial-boundary-value problem of a reaction-diffusion equation in a bounded domain with a Robin boundary condition and introduce some particular parameters to consider the non-zero flux on the boundary. This problem arises in the study of mosquito populations under the intervention of the population replacement method, where the boundary condition takes into account the inflow and outflow of individuals through the boundary. Using phase-plane analysis, the present paper studies the existence and properties of non-constant steady-state solutions depending on se…
We establish the existence and find some qualitative properties of open sets that minimize functionals of the form $ F(λ_1(Ω;β),\dots,λ_k(Ω;β))$ under measure constraint on $Ω$, where $λ_i(Ω;β)$ designates the $i$-th eigenvalue of the Laplace operator on $Ω$ with Robin boundary conditions of parameter $β>0$. Moreover, we show that minimizers of $λ_k(Ω;β)$ for $k\geq 2$ verify the conjecture $λ_k(Ω;β)=λ_{k-1}(Ω;β)$ in dimension three and more.
We study the spectral flow of Landau-Robin hamiltonians in the exterior of a compact domain with smooth boundary. This provides a method to study the spectrum of the exterior Landau-Robin hamiltonian's dependence on the choice of Robin data, even explaining the heuristics of how the spectrum of the Robin problem asymptotically tends to the spectrum of the Dirichlet problem. The main technical result concerns the continuous dependence of Landau-Robin hamiltonians on the Robin data in the gap topology. The problem can be localized to the compact boundary where the asymptotic behavior of the spec…
In this work, we are concerned with a Robin and Neumann problem with (p(x),q(x))-Laplacian. Under some appropriate conditions on the data involved in the elliptic problem, we prove the existence of solutions applying two versions of Mountain Pass theorem, Ekeland's variational principle and Lagrange multiplier rule.
The main objective of this paper is analysis of the initial-boundary value problems for the linear and semilinear time-fractional diffusion equations with a uniformly elliptic spatial differential operator of the second order and the Caputo type fractional derivative acting in the fractional Sobolev spaces. The boundary conditions are formulated in form of the homogeneous Neumann or Robin conditions. First we deal with the uniqueness and existence of the solutions to these initial-boundary value problems. Then we show a positivity property for the solutions and derive the corresponding compari…
In this paper, we obtain the upper bounds for the Hausdorff measures of nodal sets of eigenfunctions with the Robin boundary conditions, i.e., \begin{equation*} {\left\{\begin{array}{l} \triangle u+λu=0,\quad in\quad Ω,\\ u_ν+μu=0,\quad on\quad\partialΩ, \end{array} \right.} \end{equation*} where the domain $Ω\subseteq\mathbb{R}^n$, $u_ν$ means the derivative of $u$ along the outer normal direction of $\partialΩ$. We show that, if $Ω$ is bounded and analytic, and the corresponding eigenvalue $λ$ is large enough,then the measure upper bounds for the nodal sets of eigenfunctions are $C\sqrtλ$, w…
In the current work we study a nonlocal parabolic problem with Robin boundary conditions. The problem arises from the study of an idealized electrically actuated MEMS (Micro-Electro-Mechanical System) device. Initially we study the steady-state problem in one dimension and we present some numerical results regarding its bifurcation diagram. Next the $N-$dimensional nonlocal problem steady-state problem is investigated and a Pohozaev-type identity is first obtained which then facilitates derivation of an estimate of the pull-in voltage for radially symmetric domains. Later the time-dependent pr…
In this paper, we consider the exact boundary controllability and the exact boundary synchronization (by groups) for a coupled system of wave equations with coupled Robin boundary controls. Owing to the difficulty coming from the lack of regularity of the solution, we confront a bigger challenge than that in the case with Dirichlet or Neumann boundary controls. In order to overcome this difficulty, we use the regularity results of solutions to the mixed problem with Neumann boundary conditions by Lasiecka and Triggiani to get the regularity of solutions to the mixed problem with coupled Robin …
We study the long time behavior, as $t\to\infty$, of solutions of $$ \left\{ \begin{array}{ll} u_t = u_{xx} + f(u), & x>0, \ t >0,\\ u(0,t) = b u_x(0,t), & t>0,\\ u(x,0) = u_0 (x)\geqslant 0 , & x\geqslant 0, \end{array} \right. $$ where $b\geqslant 0$ and $f$ is an unbalanced bistable nonlinearity. By investigating families of initial data of the type $\{ σϕ\}_{σ>0}$, where $ϕ$ belongs to an appropriate class of nonnegative compactly supported functions, we exhibit the sharp threshold between vanishing and spreading. More specifically, there exists some value $σ^*$ such that the solution conv…
Let $Ω\subset\mathbb{R}^{N}$ ($N\geq1$) be a smooth bounded domain, $a\in C(\barΩ)$ a sign-changing function, and $0\leq q<1$. We investigate the Robin problem \[ \begin{cases} -Δu=a(x)u^{q} & \mbox{in $Ω$},\\ u\geq0 & \mbox{in $Ω$},\\ \partial_νu=αu & \mbox{on $\partial Ω$}, \end{cases} \] where $α\in\lbrack-\infty,\infty)$ and $ν$ is the unit outward normal to $\partialΩ$. Due to the lack of strong maximum principle structure, this problem may have \textit{dead core} solutions. However, for a large class of weights $a$ we recover a \textit{positivity} property when $q$ is close to $1$, which…
Condition imposed on the nonlinear terms of a nonlinear diffusion equation with {R}obin boundary condition is the main focus of this paper. The degenerate parabolic equations, such as the {S}tefan problem, the {H}ele--{S}haw problem, the porous medium equation and the fast diffusion equation, are included in this class. By characterizing this class of equations as an asymptotic limit of the {C}ahn--{H}illiard systems, the growth condition of the nonlinear term can be improved. In this paper, the existence and uniqueness of the solution are proved. From the physical view point, it is natural th…
The subject of the article is linear systems of wave equations on cosmological backgrounds with convergent asymptotics. The condition of convergence corresponds to the requirement that the second fundamental form, when suitably normalised, converges. The model examples are the Kasner solutions. The main result of the article is optimal energy estimates. However, we also derive asymptotics and demonstrate that the leading order asymptotics can be specified. It is sometimes argued that if the factors multiplying the spatial derivatives decay exponentially (for a system of wave equations), then t…
We consider a general second order linear elliptic equation in a finely perforated domain. The shapes of cavities and their distribution in the domain are arbitrary and non-periodic; they are supposed to satisfy minimal natural geometric conditions. On the boundaries of the cavities we impose either the Dirichlet or a nonlinear Robin condition; the choice of the type of the boundary condition for each cavity is arbitrary. Then we suppose that for some cavities the nonlinear Robin condition is sign-definite in certain sense. Provided such cavities and ones with the Dirichlet condition are distr…
We consider in this paper a reaction-diffusion system under a KPP hypothesis in a cylindrical domain in the presence of a shear flow. Such systems arise in predator-prey models as well as in combustion models with heat losses. Similarly to the single equation case, the existence of a minimal speed c* and of traveling front solutions for every speed c > c* has been shown both in the cases of heat losses distributed inside the domain or on the boundary. Here, we deal with the accordance between the two models by choosing heat losses inside the domain which tend to a Dirac mass located on the bou…
In this paper we investigate the one dimensional (1D) logarithmic diffusion equation with nonlinear Robin boundary conditions, namely, \[ \left\{ \begin{array}{l} \partial_t u=\partial_{xx} \log u\quad \mbox{in}\quad \left[-l,l\right]\times \left(0, \infty\right)\\ \displaystyle \partial_x u\left(\pm l, t\right)=\pm 2γu^{p}\left(\pm l, t\right), \end{array} \right. \] where $γ$ is a constant. Let $u_0>0$ be a smooth function defined on $\left[-l,l\right]$, and which satisfies the compatibility condition $$\partial_x \log u_0\left(\pm l\right)= \pm 2γu_0^{p-1}\left(\pm l\right).$$ We show that …
