Let $B_1$ be a ball of radius $r_1$ in $S^n(\Hy^n)$, and let $B_0$ be a smaller ball of radius $r_0$ such that $\bar{B_0}\subset B_1$. For $S^n$ we consider $r_1< π$. Let $u$ be a solution of the problem $-\La u =1$ in $\Om := B_1\setminus \bar{B_0}$ vanishing on the boundary. It is shown that the associated functional $J(\Om)$ is minimal if and only if the balls are concentric. It is also shown that the first Dirichlet eigenvalue of the Laplacian on $\Om$ is maximal if and only if the balls are concentric.
In this article, we focus on a doubly nonlinear nonlocal parabolic initial boundary value problem driven by the fractional $p$-Laplacian equipped with homogeneous Dirichlet boundary conditions on a domain in $\mathbb{R}^{d}$ and composed with a continuous, strictly increasing function. We establish well-posedness in $L^1$ in the sense of mild solutions, a comparison principle, and for restricted initial data we obtain that mild solutions of the inhomogeneous evolution problem are strong. We obtain $L^{q}$-$L^{\infty}$ regularity estimates for mild solutions, implying decay estimates and extend…
In this work, we employ the technique developed in arXiv:2109.11255 to prove rotational symmetry for a class of Serrin-type problems for the standard laplacian. We also discuss in some length how our strategy compares with the classical moving plane method.
In this paper, we consider a doubly nonlinear parabolic equation $ \partial _t β(u) - \nabla \cdot α(x , \nabla u) \ni f$ with the homogeneous Dirichlet boundary condition in a bounded domain, where $β: \mathbb{R} \to 2 ^{ \mathbb{R} }$ is a maximal monotone graph satisfying $0 \in β(0)$ and $ \nabla \cdot α(x , \nabla u )$ stands for a generalized $p$-Laplacian. Existence of solution to the initial boundary value problem of this equation has been investigated in an enormous number of papers for the case where single-valuedness, coerciveness, or some growth condition is imposed on $β$. However…
In this paper, we establish a new improved Sobolev inequality based on a weighted Morrey space. To be precise, there exists $C=C(n,m,s,α)>0$ such that for any $u,v \in {\dot{H}}^s(\mathbb{R}^{n})$ and for any $θ\in (\barθ,1)$, it holds that \begin{equation} \label{eq0.3} \Big( \int_{ \mathbb{R}^{n} } \frac{ |(uv)(y)|^{\frac{2^*_{s}(α)}{2} } } { |y'|^α } dy \Big)^{ \frac{1}{ 2^*_{s} (α) }} \leq C ||u||_{\dot{H}^s(\mathbb{R}^{n})}^{\fracθ{2}} ||v||_{\dot{H}^s(\mathbb{R}^{n})}^{\fracθ{2}} ||(uv)||^{\frac{1-θ}{2}}_{ L^{1,n-2s+r}(\mathbb{R}^{n},|y'|^{-r}) }, \end{equation} where $s \!\in\! (0,1)$, …
We prove existence of doubly connected V-states for the planar Euler equations which are not annuli. The proof proceeds by bifurcation from annuli at simple "eigenvalues". The bifurcated $V$-states we obtain enjoy a $m$-fold symmetry for some $m\ge 3.$ The existence of doubly connected $V$-states of strict $2$-fold symmetry remains open.
In this paper we provide a formal matched asymptotic analysis for large solutions to the Gelfand-Liouville problem in planar, doubly connected domains in the plane. Using these, we rigorously construct a good approximate solution to the problem.
This work is devoted to the study of the existence of solutions to nonlocal equations involving the fractional Laplacian. These equations have a variational structure and we find a nontrivial solution for them using the Mountain Pass Theorem. To make the nonlinear methods work, some careful analysis of the fractional spaces involved is necessary. In addition, we require rather general assumptions on the local operator. As far as we know, this result is new and represent a fractional version of a classical theorem obtained working with Laplacian equations.
In this paper, we construct time periodic doubly connected solutions for the 3D quasi-geostrophic model in the patch setting. More specifically, we prove the existence of nontrivial $m$-fold doubly connected rotating patches bifurcating from a generic doubly connected revolution shape domain with higher symmetry $m\geq m_0$ and $m_0$ is large enough. The linearized matrix operator at the equilibrium state is with variable and singular coefficients and its spectral analysis is performed via the approach devised in [27] where a suitable symmetrization has been introduced. New difficulties emerge…
This paper deals with existence of solutions to the following fractional $p$-Laplacian system of equations \begin{equation*} %\tag{$\mathcal P$}\label{MAT1} \begin{cases} (-Δ_p)^s u =|u|^{p^*_s-2}u+ \frac{γα}{p_s^*}|u|^{α-2}u|v|^β\;\;\text{in}\;Ω, (-Δ_p)^s v =|v|^{p^*_s-2}v+ \frac{γβ}{p_s^*}|v|^{β-2}v|u|^α\;\;\text{in}\;Ω, % % u,\;v\in\wsp, \end{cases} \end{equation*} where $s\in(0,1)$, $p\in(1,\infty)$ with $N>sp$, $α,\,β>1$ such that $α+β= p^*_s:=\frac{Np}{N-sp}$ and $Ω=\mathbb{R}^N$ or smooth bounded domains in $\mathbb{R}^N$. For $Ω=\mathbb{R}^N$ and $γ=1$, we show that any ground state so…
In this paper we consider rotating doubly connected vortex patches for the Euler equations in the plane. When the inner interface is an ellipse we show that the exterior interface must be a confocal ellipse. We then discuss some relations, first found by Flierl and Polvani, between the parameters of the ellipses, the velocity of rotation and the magnitude of the vorticity in the domain enclosed by the inner ellipse.
By applying implicit function theorem on contour dynamics, we prove the existence of co-rotating and travelling patch solutions for both Euler and the generalized surface quasi-geostrophic equation. The solutions obtained constitute a desingularization of points vortices when the size of patch support vanishes. In particular, solutions constructed in this paper consist of doubly connected components, which is essentially different from all known results.
In this paper, we prove the existence of doubly connected V-states for the generalized SQG equations with $α\in ]0,1[.$ They can be described by countable branches bifurcating from the annulus at some explicit "eigenvalues" related to Bessel functions of the first kind. Contrary to Euler equations \cite{H-F-M-V}, we find V-states rotating with positive and negative angular velocities. At the end of the paper we discuss some numerical experiments concerning the limiting V-states.
Motivated by recent results regarding the equivalence of the Dirichlet and Neumann problems for the Laplace operator in the case of simply connected regions, the present paper takes a step further and provides a similar equivalence between the above mentioned problems in the case of planar doubly-connected regions. The equivalence means that solving any of these problems leads by an explicit formula to a solution of the other problem. In addition, suffcient conditions guaranteeing the uniform Holder continuity of the higher order partial derivatives of the solutions to the Neumann problem are …
In this paper we consider the Liouville equation $Δu +λ^2 e^{\,u}=0$ with Dirichlet boundary conditions in a two dimensional, doubly connected domain $Ω$. We show that there exists a simple, closed curve $γ\subset Ω$ such that for a sequence $λ_n\to 0$ and a sequence of solutions $u_{n}$ it holds $\frac{u_{n}}{\log\frac{1}{λ_n}}\to H$, where $H$ is a harmonic function in $Ω\setminusγ$ and $\frac{λ_n^2}{\log\frac{1}{λ_n}}\int_Ωe^{\,u_n}\,dx\to 8πc_Ω$, where $c_Ω$ is a constant depending on the conformal class of $Ω$ only.
In this article we study the existence of weak solution, existence of ground state solution using Nehari manifold and existence of infinitely many solutions using Fountain theorem and Dual fountain theorem for a class of doubly nonlocal Kirchhoff-Choquard type equations involving the variable-order fractional $p(\cdot)-$ Laplacian operator. Here the nonlinearity does not satisfy the well known Ambrosetti-Rabinowitz type condition.
Using hyperbolic form convolution with doubly isometry-invariant kernels, the explicit expression of the inverse of the de Rham laplacian acting on m-forms in the Poincaré space is found. Also, by means of some estimates for hyperbolic singular integrals, we obtain L^p-estimates for the Riesz transforms passing from the Laplacian to other covariant derivatives, in a range of p depending on m,n. Finally, using these, it is shown that the Laplacian defines topological isomorphisms in the scale of form Sobolev spaces, for m different from n/2,(n+1)/2,(n-1)/2.
We establish the local continuity of locally bounded weak solutions (temperatures) to the doubly singular parabolic equation modeling the phase transition of a material: \[ \partial_t β(u)-Δ_p u\ni 0\quad\text{ for }\tfrac{2N}{N+1}<p<2, \] where $β$ is a maximal monotone graph with a jump at zero and $Δ_p$ is the $p$-Laplacian. Moreover, a logarithmic type modulus of continuity is quantified, which has been conjectured to be optimal.
In this contribution, we study a class of doubly nonlinear elliptic equations with bounded, merely integrable right-hand side on the whole space $\mathbb{R}^N$. The equation is driven by the fractional Laplacian $(-Δ)^{\frac{s}{2}}$ for $s\in (0,1]$ and a strongly continuous nonlinear perturbation of first order. It is well known that weak solutions are in genreral not unique in this setting. We are able to prove an $L^1$-contraction and comparison principle and to show existence and uniqueness of entropy solutions.
In this article we prove that the spectrum of the Laplacian on $k$-forms over a noncompact flat manifold is always a connected closed interval of the nonnegative real line. The proof is based on a detailed decomposition of the structure of flat manifolds.
We study the Laplacian with zero magnetic field acting on complex functions of a planar domain $Ω$, with magnetic Neumann boundary conditions. If $Ω$ is simply connected then the spectrum reduces to the spectrum of the usual Neumann Laplacian; therefore we focus on multiply connected domains bounded by convex curves and prove lower bounds for its ground state depending on the geometry and the topology of $Ω$. Besides the area, the perimeter and the diameter, the geometric invariants which play a crucial role in the estimates are the the fluxes of the potential one-form around the inner holes a…
In this paper we will prove the nodal line $N$ of the second eigenfunction of the Laplacian over some simply connected concave domain $Ω$ in $\mathbb{R}^2$ must intersect the boundary $\partialΩ$ at exactly two points.
In this paper we study left-invariant Laplacians on compact connected groups that are form-comparable perturbations of bi-invariant Laplacians. Our results show that Gaussian bounds for derivatives of heat kernels enjoyed by certain bi-invariant Laplacians hold for their form-comparable perturbations. We further show that the parabolic operators associated with such left-invariant Laplacians, in particular, with the bi-invariant Laplacians, are hypoelliptic in various senses.
We compute the Green's function for the Hodge Laplacian on the symmetric spaces M\timesΣ, where M is a simply connected n-dimensional Riemannian or Lorentzian manifold of constant curvature and Σis a simply connected Riemannian surface of constant curvature. Our approach is based on a generalization to the case of differential forms of the method of spherical means and on the use of Riesz distributions on manifolds. The radial part of the Green's function is governed by a fourth order analogue of the Heun equation.
Given a connected locally finite simplicial graph $ G$ with vertex set $V$, the combinatorial Laplacian $Δ_G \colon \R^V \to \R^V$ is defined on the space of all real-valued functions on $V$. We prove that $Δ_G$ is surjective if $G$ is infinite.
