In the last decades comparison results of Talenti type for Elliptic Problems with Dirichlet boundary conditions have been widely investigated. In this paper, we generalize the results obtained in arXiv:1909.11950 to the case of p-Laplace operator with Robin boundary conditions. The point-wise comparison, obtained in arXiv:1909.11950 only in the planar case, holds true in any dimension if p is sufficiently small.
Let $Ω\subset \mathbb{R}^n$ be an open, bounded and Lipschitz set. We consider the Poisson problem for the $p-$Laplace operator associated to $Ω$ with Robin boundary conditions. In this setting, we study the equality case in the Talenti-type comparison stated in \cite{AGM}. We prove that the equality is achieved only if $Ω$ is a ball and both the function $u$ and the right hand side $f$ of the Poisson equation are radial.
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.
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.
We compare the solutions of two Poisson problems in a spherical shell with Robin boundary conditions, one with given data, and one where the data has been cap symmetrized. When the Robin parameters are nonnegative, we show that the solution to the symmetrized problem has larger (increasing) convex means. We prove similar results on balls. We also prove a comparison principle on generalized cylinders with mixed boundary conditions (Neumann and Robin).
We study the asymptotic behavior of the solutions of a boundary value problem for the Laplace equation in a perforated domain in $\mathbb{R}^n$, $n\geq 3$, with a (nonlinear) Robin boundary condition on the boundary of the small hole. The problem we wish to consider degenerates under three aspects: 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 …
In this paper we consider PDE's problems involving the anisotropic Laplacian operator, with Robin boundary conditions. By means of Talenti techniques, widely used in the last decades, we prove a comparison result between the solutions of the above-mentioned problems and the solutions of the symmetrized ones. As a consequence of these results, a Bossel-Daners type inequality can be shown in dimension 2.
In this paper, we prove two unique continuation results for second order elliptic equations with Robin boundary conditions on $C^{1,1}$ domains. The first one is a sharp vanishing order estimate of Robin problems with Lipschitz coefficients and differentiable, sign-changing potentials. This generalizes the result for the "Robin eigenfunctions" in [26], which deals with the case with constant potentials. The second result is a unique continuation result from the boundary -- any non-trivial solution cannot vanish at infinite order from the boundary or vanish on an open subset on the boundary. Su…
Polar differentiation matrices for the Laplace equation in the disk subjected to nonhomogeneous Dirichlet, Neumann and Robin boundary conditions and the biharmonic equation subjected to nonhomogeneous Dirichlet conditions
In this paper, we establish a general weighted Hardy type inequality for the $% p-$Laplace operator with Robin boundary condition. We provide various concrete examples to illustrate our results for different weights. Furthermore, we present some Heisenberg-Pauli-Weyl type inequalities with boundary terms on balls with radius $R$ at the origin in $\mathbb{R}^n$.
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 provide a comparison result between the solutions to the torsion problem for the Hermite operator with Robin boundary conditions and the one of a suitable symmetrized problem.
In this paper, by using Schwarz rearrangement and isoperimetric inequalities, we prove comparison results for the solutions of Poisson equations on complete Riemannian manifolds with $Ric\geq (n-1)κ$, $\, κ\geq 0$, which extends the results in \cite{ANT-Talenti-a}. Furthermore, as applications of our comparison results, we obtain the Saint-Venant inequality and Bossel-Daners inequality for Robin Laplacian.
We investigate the heat kernel with Robin boundary condition and prove comparison theorems for heat kernel on geodesic balls and on minimal submanifolds. We also prove an eigenvalue comparison theorem for the first Robin eigenvalues on minimal submanifolds. This generalizes corresponding results for the Dirichlet and Neumann heat kernels.
In this paper we consider an anisotropic Robin problem driven by the $p(x)$-Laplacian and a superlinear reaction. Applying variational tools along with truncation and comparison techniques as well as critical groups, we prove that the problem has at least five nontrivial smooth solutions to be ordered and with sign information: two positive, two negative and the fifth nodal.
In this paper, we prove an upper bound for the first Robin eigenvalue of the $p$-Laplacian with a positive boundary parameter and a quantitative version of the reverse Faber-Krahn type inequality for the first Robin eigenvalue of the $p$-Laplacian with negative boundary parameter, among convex sets with prescribed perimeter. The proofs are based on a comparison argument obtained by means of inner sets, introduced by Payne, Weimberger and Polya.
In this paper we study double phase problems with nonlinear boundary condition and gradient dependence. Under quite general assumptions we prove existence results for such problems where the perturbations satisfy a suitable behavior in the origin and at infinity. Our proofs make use of variational tools, truncation techniques and comparison methods. The existence of the obtained solutions depends on the first eigenvalues of the Robin and Steklov eigenvalue problems for the $p$-Laplacian.
In this paper we prove a strong comparison principle for radially decreasing solutions $u,v\in C_{0}^{1,α}(\Bar{B_R})$ of the singular equations $-Δ_p u-\frac{1}{u^δ}=f(x)$ and $-Δ_p v-\frac{1}{v^δ}=g(x)$ in $B_R$. Here we assume that $ 1<p<2 , \; δ\in (0,1)$ and $f,g$ are continuous, radial functions such that $0 \leq f \leq g$ but $f\not \equiv g$ in $B_R.$ For the case $p>2$ a counterexample is provided where the strong comparison principle is violated. As an application of strong comparison principle, we prove a three solution theorem for p-Laplace equation and illustrate with an example.
We consider a nonlinear Robin problem driven by the sum of $p$-Laplacian and $q$-Laplacian (i.e. the $(p,q)$-equation). In the reaction there are competing effects of a singular term and a parametric perturbation $λf(z,x)$, which is Carathéodory and $(p-1)$-superlinear at $x\in\mathbb{R},$ without satisfying the Ambrosetti-Rabinowitz condition. Using variational tools, together with truncation and comparison techniques, we prove a bifurcation-type result describing the changes in the set of positive solutions as the parameter $λ>0$ varies.
This paper is devoted to the asymptotic analysis of the eigenvalues of the Laplace operator with a strong magnetic field and Robin boundary condition on a smooth planar domain and with a negative boundary parameter. We study the singular limit when the Robin parameter tends to infinity which is equivalent to a semi-classical limit involving a small positive semi-classical parameter. The main result is a comparison between the spectrum of the Robin Laplacian with an effective operator defined on the boundary of the domain via the Born-Oppenheimer approximation. More precisely, the low-lying eig…
In this paper we revisit an approach pioneered by Auchmuty to approximate solutions of the Laplace- Robin boundary value problem. We demonstrate the efficacy of this approach on a large class of non-tensorial domains, in contrast with other spectral approaches for such problems. We establish a spectral approximation theorem showing an exponential fast numerical evaluation with regards to the number of Steklov eigenfunctions used, for smooth domains and smooth boundary data. A polynomial fast numerical evaluation is observed for either non-smooth domains or non-smooth boundary data. We addition…
The networks of this -- primarily (but not exclusively) expository -- compendium are strongly connected, finite directed graphs $X$, where each oriented edge $(x,y)$ is equipped with a positive weight (conductance) $a(x,y)$. We are not assuming symmetry of this function, and in general we do not require that along with $(x,y)$, also $(y,x)$ is an edge. The weights give rise to a difference operator, the normalised version of which we consider as our Laplace operator. It is associated with a Markov chain with state space $X$. A non-empty subset of $X$ is designated as the boundary. We provide a…
We consider the first Robin eigenvalue $ł_p(M,\a)$ for the $p$-Laplacian on a compact Riemannian manifold $M$ with nonempty smooth boundary, with $\a \in \R$ being the Robin parameter. Firstly, we prove eigenvalue comparison theorems of Cheng type for $ł_p(M,\a)$. Secondly, when $\a>0$ we establish sharp lower bound of $ł_p(M,\a)$ in terms of dimension, inradius, Ricci curvature lower bound and boundary mean curvature lower bound, via comparison with an associated one-dimensional eigenvalue problem. The lower bound becomes an upper bound when $\a<0$. Our results cover corresponding comparison …
Let $\varepsilon>0$ be a small parameter. We consider the domain $Ω_\varepsilon:=Ω\setminus D_\varepsilon$, where $Ω$ is an open domain in $\mathbb{R}^n$, and $D_\varepsilon$ is a family of small balls of the radius $d_\varepsilon=o(\varepsilon)$ distributed periodically with period $\varepsilon$. Let $Δ_\varepsilon$ be the Laplace operator in $Ω_\varepsilon$ subject to the Robin condition ${\partial u\over \partial n}+γ_\varepsilon u = 0$ with $γ_\varepsilon\ge 0$ on the boundary of the holes and the Dirichlet condition on the exterior boundary. Kaizu (1985, 1989) and Brillard (1988) have sho…
Degeneracy Set Comparison Principle and Free Boundary Estimates for Hénon-type Infinity Laplace equations
School of Mathematics, Southeast University（东南大学数学学院）；Shing-Tung Yau Center of Southeast University（东南大学苏步青中心）；Southeast University（东南大学）；Shanghai Institute for Mathematics and Interdisciplinary Sciences（上海数学与交叉学科研究院）
